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no license to nse it had been obtained. We notioe these pointa, since they must be of importance to all future industrial exhibitions. The show is so creditable in itself, and so deeplyinteresting, that hitches of this sort are blameable in the extreme. Passing, howeyer, from a disagreeable topic, we may observe one announcement which is still of interest to workmen. There is to be a "Workshop Competition." "Workmen are invited to enter their names as competitors. When a sufficient number of names have been entered, arrangements will be made for a series of competitions. The council are also willing to arrange for trials of skill in other branches of handicraft, for which intending competitors should at once give in their names." Here is a novelty deserving of all support. It is, perhaps, the wisest poseible use to make of an international exhibition. And we must add that a most liberal spirit has been evinced by the Euglish 'exhibitors, who are, of course, in a majority. They saw the Italian conoourse, they acknowledged its richness and beauty, they gave it precedence. Not that it bears away the palm in point of newness, or utility, or the method of improving the ease and comfort of the life of mankind, because it betrays an ancient origin in its Roman splendour and its Tuscan grace; but that, being most lavish of art, and precious in substance, it stood, as it were, on the step of the industrial throne. As to their own contributions, in detail we must not attempt to doscribo them now. They are of an infinite variety. Amid Saturday's bell-ringing, trumpet flourishes, songs, wedding marches, coming and going of courtly crowds, crush of people, confusion of ceremonies, bewilderment of catalogues, excitemont of exhibitors, and thronging of visitors round the more popular stalls, it would have been vain to undertake a critical view of the Exhibition under its strictly industrial and artistic aspects. We hold this duty in reserve, however, and it shall not be through want of will if we fail in rendering justice to deserving competitors.

MICROSCOPICAL JOTTINGS IN TOWN

AND COUNTRY.

No. III.

I THINK we may now for a Bhort time wander from the sea-shore and pay a little attention to the land-work of microscopy. A few days since, a break occurring in my usual business hours, away from home I strolled along the line of rail.botanizing with a brother microscopist. Our "takes" were not numerous. Orchis maculata, remarkable only for its "pollen masses," we secured in large quantities, and this indeed was the real object of our stroll. My companion called my attention to clumps of the burnet (Poterium sanguisorba), and we set to work to search for the very rare brand (Xenodochus carbonari«»), which is sometimes found on that or on the " great burnet " (Sanguisorba officinalis). In this we were not successful, but we found a great number of leaves bearing another, and а very interesting fungus, the Aregma acuminatum, and also a yellow fungus, the Lecytkea Poterium. Probably as the best season for collecting microfungi is now nigh at hand (July to November), I cannot do better than chat for a while about them. Some of my readers have seen on bramble leaves iu the autumn, certain black spots on the under side, accompanied wit h reddish purple spots on the upper, and have, I dare say, at times asked themselves some such question as this, "What causes them?" Let them in the approaching autumn select a leaf so distinguished, and subject it to microscopic examination. The examiner must place the leaf on the stago of the microscope, and throw a good light on the leaf by the bull's-eye condenser. With a half-inch objective, or a good inch, he will see that the black spots are composed of small cylindrical bodies, with short Btalks arranged in small tufts. With a penknife let him remove one of these tufts, place it on a slip of glass, cover it with thin glass, and moisten with water or glycerine, and he will see that these cylindrical bodies are multiseptate or are composed of a number of little chambers or colls (four), that the terminal joint is apiculate, and that the peduncles or stalks are bulbous at the base. He may conclude that he has found the Aregmn bulboxum, of which the technical description, as given by Mr. Cooko, is "A. bulbosum—hypogenous, with a dull red stain on the upper snrtnee; spores in large tufts" (relatively large) "four septate, terminal joint apiculate;

peduncles incrassated and bulbous at the base." If a friend wish to work at the brand he can apply a little diluted nitric acid (1 in 5) to the spores, and carefully observe the effect produced. He may also test them by sulphuric acid (1 in 3) and sugar. By means of the nitric acid test he will probably find that the cylindrical body consists of an outer membrane, which is nearly transparent, and encloses a number of cells, which give the body the appearance of being jointed. There is, I think, little doubt that these cells are formed by what we may term hour-glass contraction of the primal cell-wall, and that the whole has its origin in the small yellow spores invariably found intermixed with the Aregma, and by some botanists raised into a distinct genera (Lecythea.) The burnet brand differs somewhat from that of the bramble as will be seen by the following description, which we take from Cooke's "Rust, Mould, and Mildew." "A. acuminatum— hypogenous, scattered in minute tufts; spores multiseptate, terminal joint acuminate, peduncles equal." A vast number of most interesting objects may be secured by dint of careful and patient observation by any one who cares to use his faculties to this end. I may briefly allude to a few before I pass on. Aregma mucionatum, on roses, differing from A. bulbosum in having 5 to 7 "cells," in being scattered over the leaves in minute tufts, the terminal joint being mucionate and the stalk fusiform. The raspberry brand, A. gracile, is not so common—has 7 to 9 cells, terminal joint apiculate and peduncles slender. The strawberry brand is more common, is scattered in minute tufts, is multiseptate, has terminal joint obtuse, peduncles equal, and rejoices in the name of A. obttmatum. The whole of these may be mounted dry in shallow cells, or the spores may be mounted in balsam for minute observation—glycerine is better, but is less trustworthy as regards permanence, and is infinitely more trouble. The smuts and mildews of our cornfields are worthy careful notice. The Puccinta graminis of the learned, or mildew of the unlearned (the latter, by the way, is most indiscriminately used to designate most diverse fungi), furnishes a beautiful object for moderately high power, and abounds on the leaves of grasses. Of Puccinea there are upwards of 40 species to be found on one or another of our commonest plants. "The spores are uniseptato and supported on a distinct peduncle " (Berk.). The name is derived from the Greek, and simply means "closely packed," the spores being closely arranged together. The Sedge mildew (this must not be confoimded with the smudgy sedge rust (Ustilago longissimm), the polygonum brand, the ground ivy brand, the P. compositatum, P. vinca, P. galiorum, are amongst the most common and most easily to be found. H. P.

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and on a still grander scale in the mountains ad jacent to Mont Blanc. At another, formin promontories and reefs, which stretch far^ into the ocean, and render the navigation cult and dangerous.

The principal species of granitic rocks are those called graniteproper, syenite, hyperstlienic granit-. protogine, porphyritic granite, and serpentin.:. Ordinary granite has been described iu chap, i, and therefore requires no explanation. Synii: is formed from felspar, quartz, and hornblende, s dark-vitreous substance; hypersthenic grani' consists of quartz and hypersthene, a green flu: crystal; protogine is ordinary granite, in whiol the mica is supplanted by talc; porphyria: granite is where large crystals of felspar ur mingled with ordinary granite; and serpentine t a conglomeration of all kinds of small pieces ■: mineral substance. Granite is a most dural'., rock, and is on this account much used in ели neering works. When granite appears in the form of needles, as in the Aiguille Dru (see Fig. 1¡,

THE WORLD: ITS FORMATION AND ANTIQUITY.

By Akthuk Undebhill.

(Continued from page 362.)

Chapteb IV.

I ANNOUNCED, at the conclusion of my last chapter, that I should in this number treat of the superficial or surface accumulations of the earth; but I have since thought that it would be better to commence at the other end (if I may so exprese it). I shall therefore proceed to describe the igneous or amorphous rocks, which play so distinguished a part in the drama of creation.

Igneous rocks are divided into throo groups or systems, called respectively the granitic, trappean, and volcanic; and each of these systems is again subdivided into different species of rochs, each however bearing the characteristics of its system.

Granitic rocks are easily recognizable from their appearance, being alwaye of a highly crystalline character, and from looking like a conglomeration of minute particles or grains (whence the name granite, from granum, Latin for grain) of different coloured substances. They are found in all parts of the globe, and form by far the greater portion of its bulk,—here appearing in huge masses of mountain country whose rugged surface is wholly destitute of any vegetation of a higher nature than mosses and lichens, and which elevates its summit in sharp outlines far abovo the surrounding country; there overlying tho strata in flat masses of slab-like figure; at once place found in the form of huge pikes, tors, aiguilles, or needles, as on Dartmoor,

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at Chamounix, it is evident that it could riot have taken such a shape, if it had been farmed in tho open air, and the only way by which it could have assumed such a peculiar ehape, appears tobe that it was originally injected through the strati, as yet in a soft and yielding state, and was there gradually cooled down as in a mould ; in course of time, however, the floods and rivers wore away the softer casing, leaving the hard granite in its present form, the wonder and admiration of all who see it.

Trap Rocks are generally found in the form of terraces, from which fact the name trap is derived. The trappean system embraces a great many varieties of igneous rocks, which run from the hardest basalt to the soft and earthy claystone porphyry. Basalts are nearly always found arranged in columns or pillars, end constitute some of the most curious and most beautiful tableam in nature. (See figure 2.) Staffa (a small

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island on the west of Scotland) is a mass of basalt, and is guarded on every side by these misjh'y palisades, whose height is in many places several hundreds of feet. At ono sido of the island я huge cavern burrows under tho land, forming. as it were, a natural cathedral, in which the clusters of columns supporting tho heavy arched roof of the cave, the immense size of the placo, and the architectural regularity of etructure, impress tb>beholder with far more awe and admiration than the most glorious temple built by man. Tbe Giant's Causeway, on the north coast of Ireland is another instance of basaltic formation.

Greenstones und clinkstone* are other specie-' of trap rocks; they are moro granular tlr>;) basalt, and donotoften occur iu columns. Amygdaloid and trachyte oro more oarthy and rough. and contain crystals of other minerals; trap tuj>< are a kind of igneous conglomerates; claystonr porphyries are earthy, with felspar eryatale.

All the trap rocks appear to h:iv? had a volic origin, and their' different structure arises •' the different modes and times of cooling. 'bus it has been experimentally proved that a melted mass of trap rock will, if suddenly cooled, ■ - ''i-le a columnar structure; while if, on the other hand, it is allowed to cool gradually, the result will be trap tuff, or some earthy form of the system; and thus, by varying the time and mode of cooling, it may be made to assume nearly every variety of trappean rock.

This system gives to the localities where it occurs a most pictnreeque appearance, combining a bold and hard figure with the beauties of luxuriant foliage and vegetation, nourished by the fertile soil formed from its decomposition. I may mention that the rock on which stands the town of Stirling is of the trap system.

Lastly, we come to the igneous formations of our own epoch, such as pumice stone, ashes, and scoriio, hot mud, ic. Volcanic rocks are of modern formation, and are even now formed from time to time. They, like the olden igneous systems, are injected from the interior of the globe, and in their birth display some of the grandest, though albeit the most devastating operations, of nature. ¡The eruptions of Vesuvius, for instance, are generally accompanied with frightful convulsions and earthquakes, which upheave and rupture the solid strata, forming large chasms, and finally pouring down from the crater immense streams of molten matter called lava (sometimes extending a mile in breadth and 50ft. to 6Uft. in depth), which bury in their burning embraces everything, city or cottage, river or forest, which opposes their progress. This lava, when cold, assumes an appearance somewhat similar to the earthy varieties of the trappean system. Some parts of it, however, which I may perhaps comparo to the froth or scum of a rapid river, becomes pumice-stone. Ashes and scoria; are expelled from the interior of the crater, and fall on to the earth's surface. Hot indeed, but yet in a solid state, they are often propelled to an immense distance; thus a shower of ashes expelled from the crater of Hecla, in Iceland, fell in and around the Orkney Isles, having travelled a distance of about 400 miles.

As I said before, the volcanic rocks are formed in the same way as the older igneous systems, and in consequence they exhibit many of the ваше phenomena—-at one place upheaving the crust, and rupturing the strata; at another, overlying them in flat slab-like beds; here filling up chasms caused by earthquakes—there running headlong into the sea, and forming large promontories and reefs, in the formerly deep water. By a series of such eruptions, a mountain of a conical form is reared, mostly composed of ashes and scoriao. In course of time the reef becomes overlaid with sedimentary matter deposited upon it by the water; this will in time harden, and exhibit the phenomenon so often found in the old strata of the sedimentary rocks traversed by igneous veins and masses.

Although there are now volcónos in three or four plucue in Europe, still there are remains of extinct ones all over the continent, and the earthquakes which from time to time happen in Spain and elsewhere, show that volcanic energy still exists there. In the new world, however, voléanos are plentiful, and in South America are continually in play, causing frequent earthquakes, which although not often fatal, still oblige the inhabitants to be very careful as to the mode of building their houses. Lara etreams move however very elowly, and cool very slowly too ; so much so, indeed, that it sometimes takes several years before a lava stream cools sufficiently to allow of solidification. The pace is very slow—sometimes only a few yards perday. The surface cools first, thus forming a huge mineral pipe, through which the fiery flood passes. I have now reviewed the igneous formations, and in the next chapter shall commence with the earliest stratified rocks.

(Го be continued.)

THE FLUTE AND ITS VICISSITUDES.

Вт Sable.

"Nothing extenuate, nor anght set down in nuUioe."

Intboduction.

AS I follow the flute in its fortunée, and offer illustrations of the various changes through which it has passed during the last fifty years, baring no desire to provoke, I ehall take no part in controversy. I am much indebted both to the Editor and contributors of this truly valuable periodical; and I would, in my poor way, offer

some slight acknowledgment for their kindness and courtesy. Among the most distinguished of those who shed the blight light of science on its pages stands " P. R. A. S.," and I wish, like him, to preserve my nom de plume, and thus I trust to exonerate myself from any motives of ambition— indeed, that is a long way behind me now. To interest those who are able to use and eujoy this elegant instrument, and perchance to guide beginners past some of those self-created difficulties into which they are so liable to fall, often from attempting to accomplish, in a few months, that which asks the perseverance and discipline of years—and to pass some wearisome hours usefully —this is the motive which inducee me to write. Enough that the controversial war did long and hotly wage to the vexation and perplexity of players, and to the cost of many talented and eminent men, who sacrificed something even more precious than gold. All honour to each one of them! Once upon a time the flute was a very bad instrument; now it is as good as any one can reasonably desire; from its peculiarly intractable nature, it will never be more perfect than it is, and it must be a man's own fault, or rather the fault of his pocket, if be does not possess a really fino flute. As I cast a backward look at the hist years of my life devoted to its study, I cannot help contrasting tho flute of 1830 with the flute "of the period," and as I do 60, I offer шу grateful thanks individually to those who have expended time, patience, talent, and money to make it what it is.

The flute is as old as the—no, not quito as venerable as the hills, but it is as old as the Pyre mills. Some cane flutes were discovered in one of them, having, it is said, eight finger-holes, ami although the person who found them could nut tell how they were sounded, I dare say some of our English mechanics would readily have solved the problem, and made them discourse an sweet, if not elaborate music; for the poorest of the flute's relations has within itself the very genu of melody. The following circumstance will vouch for what I say :—Some time ago, while on a visit in , I rambled on a July afternoon until I found myself in a shady lane, and 1 found, too, I had lout my way. A cottage stood at no great distance, and I bent my steps thither to enquiro my road, and to ask for a little water. for I was very thirsty. A comely woman, with very bright black eyes and a very white cap. civilly offered me a seat, saying, "an' kindly welcomo, sir." During her absence, I cast a look about the tidy room, and on an antiquated bureau I saw lying—yes, a flute. It was a poor little thing, but it was a flute for "a' that." 1 took it up, it might have cost a shilling; Sn roughly was it made that I was fain to take out my penknife and trim off the ragged fibres arouni I the inside of the embouchure. I put it to my lips, and by altering the direction of the stream of breath for nearly every note, I managed to elicit from it the two hist phrases of Rossini's "La Carita " with a sweetness that surprised me. I may mention that it had a D sharp key, but it was on the open-koyed system, and refused to remain shut. I replaced the little flute, and began to be impatient for the return of my hostess. Soon she entered with a glass of milk, saying as she handed it to me, with true feminine grace and kindness, " I was hoping as you'd play again, sir. I do love the flute, it's a sound of itself. I wish our Charley had a flute like to that one, but he's going to get a better one than he have after hops are gathered." I assured the good dame I had played on tho little flute and no other, and the look of surprise with which she regarded me "was worth a small annuity" to witness. She was right, the flute is a sound of itself; it stands quite alone, the "strings," "reeds," and "brass" ore all well supported by their own proper bass; yet the flute has ever maintained its place in the orchestra becauso its valuable services could not be dispensed with. The flute has been ruthlessly and of necessity somewhat unfairly handled : no other instrument if we except the pianoforte has undergone such пшl -il ii m ; but as the virginal, spinet, clavichord, and harpsichord gradually passed into the present pianoforte, the keyboard remained unaltered, no perplexity or difficulty was here presented to tho performer. Its compass was increased and the touch and quality of tone improved, and the player was enabled to get more expression from his finger ends. But in the flute the natural progression of the diatonic scale was broken, and it varied with every new form of the instrument; new fingerings were introduced, a difficulty was removed from one region to generate two or more

in another, and the touchstone of practical experience proved that theory and experiments had placed the flute in a still more unsatisfactory condition. Is it then any wonder that it should have been regarded with a suspicion amounting almost to dislike? That it has not of late years met its once cordial reception in the orchestra and drawing-room is not surprising. The promise of a flute accompaniment in the latter bas seldom failed to provoke a quiet smile, the " twilight of a smile," often sarcastic, and unpleasant allusions have been made to the lugubrious performances of Dick Swiveller, or a ruler or a stick has been seized by the "funny man" of the party, and the conventional bowing, ludicrous pigeonlike attitude assumed. Beginners should beware of this, it is a rock on which many split, it is fatal to tone and tune, and produces that wheezy sound, aud that alternate elevation and depression of pitch so unendurable to refined nud cultivated ears. In such u position it is impossible to intone correctly with flute or voice, because the windpipe is bent nud the air cannot равв freely, and the lungs are embarrassed. 1 know from experience in many instances how easily pupils may ibift hito this ungraceful and ruinous habit, and the toil it gires them to free themselves from it. "Many a man" says Rockstro. '' after much patient toil has laid ¡do his flute disheartened, and with disgust, thinking there was nn inherent defect in himself or his instrument, when all the while the bar to progress lay in some bad habit which a timely hint from a master would have saved him from contracting." It ie impossible to articulate perfectly in tune with a bad flute, yet a well-disciplined lip can cover many defects; and it is equally possible to blow a good flute out of tune. Tone, style, and execution are matters of study, patience, and example; but intratiuii, the perception and interpretation of the great masters, are more or less individual gifts which proclaim the true artist. When I took up the flute as a student (a long time ago) I had no one to oppose my own sweet will: I studied as I liked, and consequently I had a vast deal to unlearn. Music and painting are twin sisters, before one can be a master one mutt have n master. However, to save others from similar vexations, I will offer a few brief hints on the attitude of the player, the position of the instrument and the formation of the embouchure in my next article, aud I trust they will be received in the same spirit in which I put them forth. I close this paper with the drawing of a once rather popular instrument, the flatito dolce or flutc-u-boe. It was neither more nor lees than an incapable flageolet, with an imperfect register, but a pure and vocal sound. Its extinction I believe has been lamented by some musicians. It is figured in the descriptive catalogue of M. Carl Engel, and is amongst the musical curioeities in the South Kensington Musenm. It is made of tortoiseshell, and is stated to have belonged to Rossini. It is worthy of remark that its body is cylindrical and not conical. (To be continued.)

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II.—Principle of action. A body M, Fig. 77, laid on the length А В of the plane is acted upon by gravitation (J along the line M Q. The force of gravitation can be decomposed into two: one M О, perpendicular to the length Л В ; the other, M R parallel to A B. Now M О, or the "normal," as it is called, has no effect whatever as to the motion of M down the piano; heneo M R is the whole force to bo considered. Therefore a force M В,1, equal and opposite to M R, will keep M at rest.

The principle of the inclined plane is, then: that the length of the piano opposes part of the force acting on the body, leaving only a certain quantity to be balanced by the power P. и —Conditions of equilibrium. They are all

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contained in one small formula ту

sin a

In

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COB <-.

this formula P, Fig. 78, is the power exerted;
its direction is along
the line N P, but for
the moment no in-
tensity is marked by
length of line ; W the
weight of the body;
a the angle of the
plane, ß the angle of
the rope, with tho
"length "AB. The
proof of this formula
is simple, and just
represents the amount
of knowledge sup-
posed in this series.
Construction. Tho
forces P and Q not
being parallel act upon
the body at a point N.
Draw N O1 perpen-
dicular to A B, i.e.,
"normal" to the
"length" of tho plane.

As Q represents in w r.

direction and intensity the action of gravitation, and as the direction of the power and the normal are determined, a parallelogram completed gives the representative intensity-length of the normal and power. Draw, therefore, Q 0 parallel to N P, and О P parallel to N. Q. The gravitation and weight being ono and the same thing, the power and weight are represented by the lines N P and N Q. Proof of formula. The lines being proportional to the forces

L N p

W = NQ"

Sines and Hides opposite being proportional, it

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follows that

P
W

sinNOP

sinNOQ-
Substituting tho alternate (equal) angles,
P sin Q N 0
W = sin P NO'
But Q N 0 = a (similar triangles о N 6 and
ВАС).

P sin я

■•'W = sin P N O' Now sin P N 0 = cos P N D (D N drawn perpendicular to 0 N)

P sin a

'•'W = cos PNU'
Finally P N D = g (N D is parallel to A B).
P sin a

'•"W = coTg'

With this formula all questions possible on the inclined plane can be answered, by resolving the equation for tho unknown quantity. Thus, W sin a P cos g

P = • ^-. W = , &c. The con

cos ь вш a

ditions of equilibrium, without the help of simple trigonometry, must be studied in a sot of different cases, each having its own formula and special train of reasoning; with the formula just established, all possible cases are included in three or fonr suppositions. The simple evolution of these will show what a quantity of knowledge is securely stored up by remembering the short

P formula rn =

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sin a

r Suppose:

o,—i.e., the cord forms no angle, is
parallel to А В (fig. 77) cos g
= radius = А В tho length of
the plane. Sin a in this sup-
position may be replaced by the
height (H) of the plane. Hence
the common formula (c). Sin a
is always the height, but H can

never stand for sin a unless when cos g is in a
line like H.

(2.) S = «—¿.f., the cords form some angle
(fig.78) represented by the
indefinite quantity n. Cos g
may be represented by 1 — m.
m indicates the diminution of
of the valuo of cos g. As
none of the elements of the plane enter into this
supposition, an ordinary formnla is not possible.
Bnt the value of 1 - m (cos g) is at onco found when
absolute value is given to n. From the inspec-
tion of the formula in its general shape, it is evi-
dent that as g increases P must increase (the di-
visor cos g decreasing) until

(3.) Cos g = sin a—i.e., when the power acts
perpendicular to the hori-
_ = 1 zon, then P must be equal

to the weight. That the
power is acting along a per-
,„ ЬоЛг.оп when cos g = sin a is
easily shown. Draw*0 V (tig. 79) parallel to A
С the base, which is always taken as horizontal.
Now the angle В О V = В А С or a' = a. But
bv the supposition cos g = вш a', therefore the
a- 'le P 0 V must bo equal 90°. For the equa-
lity cos t = sin a', is only possible when V OP
is a right angle, or when P is perpendicular to the
base. That the plane is of no service in this case
experience is enough to teach; but there is a
great satisfaction in finding it so clearly indicated
by the formula : P = W.

Hence cos g = cos a. But cos
a = A C, the base (fig. 80).
Here therefore again the height
may be used instead of sin a.
This is another of the ordinary
laws P : W :: H: В. The com-
mon form in which this case is
presented shows the power (the
cord) parallel to the base, A 0. But the formula
proves that the law is general; therefore the
force may be applied in any other direction ш
which the condition ( g = a) is observed. Hence,
the force may be applied along the line P' N,
which can be so drawn that the angle P F В (/3)

"This being the first of the machines, it will be
very well to state, in a marked way, their object,
and to show how in the "present " machine the
formula indicates tho conditions for tho better
attaining of that object. In the plainest terms,
the general object of these machines is to pro-
duce equilibrium with the smallest possible power.
Hence the formula which gives P the least value.
indicates the most favourable conditions of equi-
librium. There is something unusual to a be-
ginner in this view of a formula which, if not
minded, may suggest a wrong interpretation of the
results. Therefore the least being the best, it is
evident from inspection of the formula I., that
case (1) gives the greatest advantage; for in that
case cos g is the largest possible divisor. II. Case
(2) comes next as long as it keeps from (4) g = o,
which is the worst of ordinary conditions. III.
After g = a, at each increase of angle there is a
nearer approach to the machine being useless.
These conclusions suppose that the plane elements
do not change. How they should change to decrease
P is also shown by the formula I. The smaUer
sin a (the smaller the anglo of the plane), the
smaller will be the quotient = P. Here is the
practical experience: that the less the height of
the plane the better. II. The greater cos g
(the longer the plane), the greater the divisor, and
consequently, &c. This falls back into the first,
in practice: the height remaining unchanged, the
longer the plane the less the angle. III. No
lengthening of the Ьаве will give- increase of
power unless the anglo be changed, for the ref-
lation of sin a and cos. a remains the same.
These elements of the inclined plane serve to in-
troduce a most important consideration regarding
all machines—namely, what is gained inpower is
lost in time. Thus here, when the planeis longer
with the same height, there is a gain of power,
but a loss in the time required to move the body
to the same height. Suppose the plane 600 ft.
long and 10 ft. in height, the body has been only
raised 10 ft. after tho time spent in moving over
100 ft. Were the plane but 30 ft. long, with the
same height, there would bo a loss of power (P
should be greater), but there would be a gam in
time. This gain or loss snggeste the announce-
ment, even now, of a principle most important m
practical mechanics, which is usually explained
first, in another place. Still as a rigid demon-
stration of it cannot be given even there, its
simple explanation is well obtained by means of
the inclined plane. Such is its value that it is

known under a special name—the principle a virtual velocities. The word " velocity" веет to requin- force in action, yet it is not so. Thi principle is more easily understood by suppo=uq action for the moment, but it applies in all u fulness to equilibrium, or more strictly is onr. for forces not allowed to act. Hence in the namt, the word " i irtual." i. е., the velocity which tin force could produce if allowed to act. The ei*. mination of what takes place when a weight i.i raised along an inclined plane shows, in a smy and comprehensive manner, the main truth «e tained in the principle of " virtual velocities." body on an inclined plane has a certain weigkl When it has arrived at the top it has moved w-J a certain velocity through a given perpendicniJ space. On the other hand the power which rabi it has a certain amount of force, which could I represented by a weight. When the body Ы been drawn up, the power, too, has acted wit certain velocitv, and through a given f-pj Take case (1), and let А В = 100 ft., А С = W H 60-10 . _ ,. W = 60, P = -j- = щГ = 5- Hcrt" power 5 has to move through а регреиШешspace of 100, as it must draw the weight over tilength A B. In the mean time the weight |i only moves through the perpendicular spse А С = 10. Now multiplying the power by hi врасе (50 x 10), and the weight by its spart (50 x 10), there results an equality (1) 50(1 = 500; or (2), 500 - 500 = 0. Substituting L these equations the possibility for the actualttf (the posse for the esse) of the result, the principb of virtual velocities signifies: that in allmaelam there mu.it be equality between tiie power, mult: plied by its velocity, and the weight (or wort to be done) multiplied by its velocity, hi the fractional form it ransl= —; tie power is to tie

W v work inversely as the possible velocities (r = velocity of P, V velocity of W). Heme (2) P i> X W v' = 0. This last formula will шаке intelligible the general expression of '■ virtual velocities" to be met with in books—Ï (p x ») = «— i.e., the algebraical sum of all the Yraantf (forces of power or weight-work) mmüpued »J their respective velocities is equal to zero. Иш expression is in nothing essentially different from the formula (2). Instead of P v is the sum of all the forces which may constitute the sum ol r <

pv + p'V + Instead of to v »*

sum of its constituent forces, which arc alsoi>dered bv the letter {p) with different indxcei. 1* letter lp) is used for all, both power and *af elements, to express pressure—forces not in*** —equilibrium. .

IV.—Development. Inclined plane леев. «»* from one point to another raised above it. Tb; are no more than very long inclined plan« £ small angle. Here occurs a technical term toexplained—for example : " the roadnsesl «LThat si-nines that at the end of each U. <■plane is one higher than at the end of the ¡.receding 12. Suppose 1 = a foot, 1 m 12 mgw*» that the road is one foot higher at the end' every 12 feet. The same is the meaning of l » 20 30 &c. From the principle of the mclinedplm^ it is evident that the great object in the constiw; tion of roads is to make the incline or " S^f^f) as they are now called, as gentle as possible,.u>■. a as small as circumstances will permit. W gru> inclines there are some wonderful «^P/** ¿°¿ temporary railway over Mont Cew» ^»^ ever vet attempted with success by a '«^T The "gradients" of the zig-zag ^chnedpj.. are 1 i.. 12 and 1 in 13. The aids to the ,gm* the steam will be brought m i«te■<*«

the steam will be brougui ш "£'' Gnat>b-
principles. The railway over the Western fata
in India had before the Mont Cents the gr«
incline. Crossing the Western Ghauts, ш
miles there is an ascent of 1,800ft.

The raising of great weights from one lev another is the second use of the Mohm*£ Blocks of stone from thequarry and jg¿ (V up into the dockyard are the chief pracuui amples on a large scale; on a small scale«.*»^ are without number, and too °b«oubto £4 special mention. On the Morris canal, New a U.S.A., there is an example of a novel useo , inclined plane. Instead of a sen es of w , this canal has a series of twenty-three '» planes, varying i" height from 85rt. TM $, The boats (carrying some 70 tous of oo*l drawn to the top of the plane by a ropei «» ^.,. a drum 12ft. in diameter, ana worked^ ^ machinery. There is found to be a g"*';, lwl¡ i of time as compared to lock lifting.

I » Engineering, VoL vi. P-10"

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Г Snb-application I.—The wedge I., an inclined plane forced under a weight, has the same action as if the weight were drawn up along the plane. Some of the most ordinary uses of a wedge show that it is no more than an inclined plane driven by blows under some weight; or in more general terms, at an angle to some forces to be overcome. II. The wedge, when used for lifting, is a single inclined plane, the force is applied parallel to the base, by blows on what is called the "back" of the wedge (the "height" of the plane). For all effects of separation the wedge is a double inclined plane III. Theoretically, therefore, the wedge should bo reducible to the formula of the inclined plane. Bat practically, the wedge requires attention to forces, the power of which can only be calculated more or less approximately. Thus the power (P) is usually exerted by a blow, the exact value of which it is very difficult to estimate. Then there enter elements, some of which it is almost impossible to bring under the eontrol of, mathematical fornrnlie. A good notion of the difficulty may be formed by reading in the English Mechanic, April 15, the article on the wedge, in the valuable series of papers on Friction, by Mr. C. Draper.

(To be continued.)

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A New Star Allai. By R. А. Рпостоп, B.A., F.It.A.S. London: Longmans, Green, & Co. 1870.

IF we may credit Clemens Alexandrinns, the first division of the heavens into constellations was made by Chiron some 1420 years before the Christian era. Pliny, however, speaks of astronomical observations found on burnt bricks f" coctilibm latereuli ") among the Babylonians, ascending more than 2200 years B.C., while Alexander sent to Aristotle from Babylon а sot of Buch observations reaching 1900 years back; and it is asserted that the Chinese and Indian observations extend back to a more remote peri od still. Reference in the Bible too, ав in Job xxxviii., 31, 82, to individual stars and groups, indicates that various Btellar configurations had been remarked, at a very early age. Knowing then, as we do, that hieroglyphics, or pictures, have always preceded writing, as the means of record employed by nations in their primitive emersion from barbarism, it seems only natural to believe that figures of star groups, of some sort, must date from the very hoariest antiquity. Quitting, however, the region of conjecture for one of ascertained

Siientijtc Bcvuw, Vol. ii., p. 80S.

fact, we find Ptolemy speaking of Polaris as " the star at the end of the little Bear's tail," Regulas as " the Lion's heart," &c.; and it is curious that this mode of description actually survived to the time of Halley, our second English Astronomer Royal; albeit, nearly 180 years before, that very voluminous writer, Piecolomini, in a work on the fixed stars and the sphere, bad designated the stars of each constellation by letters, ami even inserted some rude maps of them thus distinguished. Some 50 years later than Piecolomini, Bayer gained what has been called "cheap immortality," by a revival of this idea, and attached to the stars in each constellation, in the order of their brightness, the letters of the Greek alphabet, commencing with the capitals, going on to the small letters; and finally, on exhausting them, using Roman type for the remaining components of the constellation. The recognised constellation makers are Aratns and Ptolemy, and (perhaps) Bayer, Hevelius, and Lacaille. Others, including Halley, have made confusion worse confounded by poking in foolish little aeterisms into the midst of a scheme already most exceedingly involved; and by adding where they might very well have subtracted, only made bad worse. There can be no doubt of the nse of constellation-figures as a kind of memoria technica for those whose object it is to acquire a naked-eye acquaintance with the heavens; but those of Ptolemy were sufficiently confused, and interlaced in a sufficiently perplexing way, without requiring modern help to still further complicate the artificial divisions of the face of the visible sky.

The first stellar maps, worthy of the name, were those of Bayer, which appeared in 1603. Flarasteed's "Celestial Atlas," published in 1729, consists of 28 charts, and was the principal one employed by astronomers dnring the last century. Bode's atlas, published in Berlin in 1805, contains between 17,000 and 18,000 stars. Harding's, in 27 charts, gives more than 50,000. Since then Bessel's zones and Argelonder's "Urnuometria Nova" have appeared. Weisse's reductions of Bessel's zones contain nearly 32,000 stars, ascending to the 9th magnitude. The chief excellence of Argelander's maps consists in their containing a thoroughly revised scale of magnitude of stars, they having been by no means previously all classed exactly according to their apparent brightness. A large atlas has also been brought out in Paris, by M. Chas. Dien, within the last few years. A great drawback to the popular use of the majority of the works published during the present century is found in the fact that the stars in them are dotted down without note or symbol, and suggest the idea that a brush full of Indian ink has been rapidly drawn across a small tooth comb, while the latter has been held over a sheet of white paper. From 1830 up to the present year essentially the maps of the stars which have been in use for ordinary purposes in England are those of the Society for the Diffusion of Useful Knowledge, and, with certain reservations and exceptions, very excellent they are: that, however, they are defective in more than one point of detail everyone who has ever used them will admit. A compilation from them came out about the year 1842 under the title of "Middleton's Celestial Atlas;" this contains a most valuable adjunct in tho shape of corresponding blank maps, presenting the stars white upon u dark ground, as we see them in the heavens. A very picturesque " Atlas of Astronomy " too was produced (under the editorship of Mr. J. R. Hind, F.R.S., superintendent of the " Nautical Almanac ") some fourteen years ago by the great Scotcli geographer, Mr. A. K. Johnston. It is beautifully printed and got up. The most glaring and obvious fault in all these atlases, without any exception, is the great distortion which exists when we recede from the very middle of the maps. To take the Useful Knowledge Society's maps as a typical example— they are drawn on the six interior surfaces of a cube, on what is known as the gnomonic projection. A popular notion of tliis construction may be gained if we conceive the stars to be accurately delineated upon the internal surface of a sphere, this sphere to be placed inside of a cube or box whose sides, lid, and bottom are tangent planes to the sphere (or, as one might say, the cube exactly holds it), and the eye being placed at the centre of the sphere, lines drawn from it through the various stars and produced until they pass through the faces of the cnbo, then the places of the stars will be marked down upon the inside of those faces at the points where our imaginary lines pass through them, and upon dissecting the cube we shall have six square maps representing the whole of the visible heavens, It needs a very rudimen

tary acquaintance indeed with mathematics to enable any one to see that while tho configuration of the stars will correspond pretty accurately where the cube touches the sphere, i. е., in the middle of the maps, when wo get up to the boundaries of the maps, or to the edges and corners of our imaginary box, the distortion becomes unmanageable, and well known stellar configurations are twisted and cramped up so as to be unrecognisable.

The most prominent characteristic of the admirable atlas tho title of which heads this notice, is the manner in which this gravo defect is obviated, by the assumption that the spheio is containednot in a cnbe, but in a regular dodecahedron, thus dividing it into twelve equal and similar spherical pentagons, and by the employment of what is called the equidistant projection. For a detailed account of the method adopted we must refer to the excellent and readable introduction to tho atlas itself. In practice the pentagonal boundaries of the maps are replaced by their circumscribing circles, во that the respective maps overlap, not at their extreme edges, bnt weU within the area delineated. It is, of course, physically impossible to reproduce a spherical вигface with absolute accuracy on a plane; but the amount of distortion in the particular species of projection adopted is so very nearly insensible as, for practical purposes, to vanish. It is only needful to compare the twelve circular maps which constitute the atlas with the two key maps prefixed to them, and drawn on the gnomonic projection, to see the enormous superiority of the former.

The scale of the maps being that of a 20in. globe, affords ample space for the delineation of detail, and tho consequence is that many thousands of objects are charted without the slightest confnsion. Some rude notion of tho legibility (if we may employ that word) of the maps may be gathered from tho statement that on map 2, a and ß Andromedœ are 2-5in. apart. These maps have been very beautifully reproduced in photo-lithography by Mr. A. Brothers, F.R.A.S., to whom high praise is due for Ids share in tho matter. Mr. Proctor has not, however, confined himself to the manifest improvements we have indicated; a very enrsory inspection of these maps will reveal many others. He has discarded, the rays which previous uranographera have always put round the images of tho stars, and represents them as circular discs. A very simple and ingenious method is employed, too, in indicating the various magnitudes, and we may add that the adoption of Sir John Herschel's photometric scale enables him to indicate the relative brightness of the leading Btars, an exceeding help to beginners, who may well have been often puzzled to sec the smaller of two stars in a map the brighter one in the sky.

In the maps themselves, the constellation figures are abandoned. Deferring, however, to popular prejudice, our author has retained them in the index maps, where we may see a brown bull with bright yellow horns butting at Auriga, who is appropriately got up in pea-green.

The names of the constellations af e put on small labels in those parts the most free from Btars, and Colonel Strange's capital forms of numerals are used throughout. The hour circles and parallels of declination are marked in continuous lines—the lines of latitude and longitude in dotted ones. A very simple arrangement of arrowB indicates the processional motion for 100 years, so that the maps serve, and will serve, for any epoch. The symbols employed to indicate nebula?, clusters, binary and multiple stars, &c, are all of a very simple description, and the whole of the detail of co-ordinates and nomenclature, while perfectly sharp and distinct, is yet во relatively faint as not to distract tho eye from tho various discs representing tho stars themselves.

Probably a difference of opinion may exist as to the expediency of tho alteration made in the names of some of the constellations by Mr. Proctor. The chief objection to such alteration would seem to be that it throws a needless difficulty in the way of the beginner, who may be searching for somo object from its name in a catalogue. For example, tho incipient possessor oí a "Nautical Almanac" might not quite reeognizo 12 Canum Venaticorum in " a Catulorum," nor Procyon in « Felis. And while mentioning this last asterism by the way, we remember that some former starcollector actually did insert a constellation Felis, somewhere under Hydra. We may add too, that if it be needful to alter Ursa Minor at all, we would indicate Ursula as more suggestive of its old appellation, than merely "Minor," a name equally pertaining to Leo Minor and Canada Minor, both of which figure here under a changed nomenclature.

The atlas is strongly but handsomely bound, is of the handy size for reference, and is sufficiently elegant in its external appearance to render it an ornament to any drawing-room table. For the special purposes for which it was designed it is certainly not only unsurpassed but unapproached by anything which has hitherto appeared; and, while it will doubtless soon be found in every fixed observatory in the kingdom, its peculiar adaptation to the wants of the astronomical student« unprovided with graduated instruments, will commend it to that large and increasing class of observers who merely possess moderate telescopes on ordinary stands. They may accept our assurance that in no other way can they acquire a familiarity with the face of the celestial vault so easily and pleasantly as through the medium of the work of whose leading features we have endeavored to give some indication.

On the ilanu/acture of Beet-root Sugar in Eng land and Ireland. By William Crookks, F.R.S., &c, Editor of the Cltemical News. London: Longmans.

The manufacture of sugar from beets is an industry which bids fair to become established in the United Kingdom; and from a conviction of its great national importance Mr. Crookes has been induced to prepare this elaborate work for the press. In the year 1717 Marggraf, a Berlin apothecary, made known at the Royal Prussian Academy of Science the fact that beetroots contained sugar identical with that derived ¡rom the sugar-cane ; and although his proposals to extract it on a largo scale, owing to a variety of causes, did not then meet with any success, within a century after its announcement his discovery became an important industry, and the extraction of sugar from beet is now one of the recognized manufactures, not only of Germany butof France, Austria, Russia, and Belgium. The original plant from which the beet of agriculture was derived grows wild on the coasts of the Mediterranean, in Spain, Dalmatia, and some parts of France. It is an unsightly bi-annual plant, known to botanists as the Beta maritima ; and the common red beet is a variety of the same genus. The percentage of sugar contained in beet-roots varies considerably with the nature of the soil, the method of culture, and the kind of seed used. In a number of analyses made by Professor Voelckcr, he found the amount of sugar to vary from 3-62 per cent, in a white beet, to 13-19 in a red, the latter manured with London sewage. Professor Church has found as much as 1317 per cent, in Carter's prize nursery sugar beet, and Messrs. Vilmorin, of Paris, are reported as having seed which will produce roots giving 16 per cent, sugar. The chief requisites in soil upon which this crop is intended to be raised are a sufficient depth and ready penetrability by the plant. A good friable loam, and all soils which grow potatoes in perfection, are the most eligible for the growth of beets. Clay lands will be found equally well adapted, if the soil is judiciously prepared. It would, of course, be bad farming to cultivate the best two years in succession on the same piece of ground; but it forms an excellent rotation crop with wheat, clover, and oats, or with wheat, flax, and turnips, all of which, according to Mr. Barnchson, are more abundant when beet forms the alternate crop. That this is especially the case with wheat all who have made the experiment can testify. One peculiarity of the beet, and one of its great advantages to farmers, is that, added to the fact that its leaves make an excellent fodder for cattle, the pulp, after extraction of sugar, makes a valuable feed for stock, and according to Prof. Voelcker, although he speaks with some reservation, one ton of the pulp as it comes from the presses contains as much nutriment which cattle can assimilate as 1J ton of the roots before the sugar is extracted, or as much as 2 tons of common mangolds. "In Belgium fattening beasts are sometimes fed upon nothing else but the refuse pulp of sugar manufactories." "A fair allowance of beau-meal or cotton-cake, and as much pulp as the cows will eat, produces both abundance of milk, and milk of good quality." Pigs will also do well upon it, if they likewise receive barley or pea-meal, or a mixture of both.

Having seen that the cultivation of the beet must be profitable, we will proceed with Mr. Crookes to inquire into the pecuniary aspect of the manufacture of sugar and spirit from beets. Of these, the latter seems the more profitable, but the distilling should be connected with the farm

ing, and carried on simultaneously. Thus, a Mr. Robert Campbell has created a large beet-root distillery on his estate at Buscot Park. This agricultural establishment comprises about 6,000 aeree of land, and the distillery supplies sufficient refuse to feed 12,000 sheep and 2,500 oxen. Mr. Campbell has obtained on the first trial about 40 tons of beetroot to the acre, with from 10 to 12 per cent, of sugar. A French manufacturer, M. cum. ponnois, has succeeded in extracting the whole of the sugar contained in the beet, and converting it into alcohol, and in keeping in the residuary product the whole of the alimentary principles which the beet contained. The value of the spirit extracted is not the only return for the original outlay; for you the land belonging to the distilleries of which Mr. Crookes gives statistics, the yield of wheat was increased from 19 hectolitres to 27 i per hectare; and whilst these farms were found capable of rearing only 25.386, and fattening G,l,i55, head of cattle, they now rear 51,449, and fatten 46,656 head, besides giving employment to the vastly increased number of workpeople.

Mr. Crookes gives the following estimate of the expenses of a sugar manufactory, the first cost of which he puts at £13,157, and considers capable of producing at least 1,200,0001b. of raw sugar. The amount required for labour and material would be £12,620, which, together with taxes, insurance, and interest on capital, would make a total expenditure of £13,980. The realization from this outlay should, reckoning only 8 per cent, yield of sugar, reach £20,470, thus leaving £6,490 as the profit! Mr. Barucluson however, taking the yield of sugar at 6 J per cent., estimates the return at upwards of 24 per cent, on an outlay of £30,630; but if the roots yield 8 per cent, of sugar, the extraordinary profit of 48 per cent, will be the result. Even taking these statements cum grano talis, it is sufficiently evident that profitable employment can bo found for money lying idle; and we can confidently recommend this book to any capitalist in search of a remunerative speculation. Mr. Crookes has spared no pains to obtain all the most recent information; and his work contains, we imagine, the com pletest account of the manufacture of beet-root sugar ever published.

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IMPROVEMENTS in experimental methods have, during our epoch, kept pace with improvements in analytic methods. Formerly experiments were made upon pieces of small dimensions; and the constants which were determined were applied to the same materials, whether employed Let the smallest or the greatest works. The old experimenters followed this method: Buffon with wood, Rondelet and Remiie with materials employed in masonry, Barlow and Tredgold with iron and steel, Vicat with mortars. They had in view generally the determination of four quantities—the specific gravity, which enters into the calculation of the proper loads, the coefficient of elasticity, the limit of elasticity, and the breaking weight. A fifth remained to be determined, that is, the limit of the load for actual construction consistent with durability. These experiments, repeated by a great number of observers, under the most diverse conditions, have thrown great light upon engineering art. The tables of Genieys, in the section on the resistance of materials, present an excellent rename in a succinct form.

The problem of the reaction upon pieces at their supports which leads to particular cases, has been studied experimentally; and Rondelet, in his "V. de Batir," reviewing all the facts collected by his predecessors with reference to the compression of wood, has given a practical rule which indicates the successive reductions to which the limit of load should be subjected, as the ratio of the length of the piece to its least transverse dimension is increased. It follows from this rule that a piece twenty-four times as long as the side of its square section upon which it stands, will sustain only half the pressure which it would if reduced to the form of a cube. This subject has been taken up and perfected by the English engineers, whose experiments we shall now notice.

Among these experiments the most numerous and remarkable are those of Mr. Hodgkinson; the results of which were published in 1846, and which were translated in 1855 by M. l'irel in " Les Anuales des Ponts et Chanssees." The experiments of Hodgkinson were intended to determine with precision the extreme resistance of steel to traction, to fioxiwn, and to transverse strains, and to deduce from the observed phenomena the best form to give to solids made of this material. The old observers,

always preoccupied with the use of formula? deduee-J from the theory of flexion, did not attach great importance to the phenomena attendant upon«¡ rupture, and the majority neglected to extend ttuir experiments beyond the limit of elasticity where the formulas cease to be applicable. The observations of Hodgkinson sensibly enlarged the field at the old experiments.

About this time the extended use of metal in construction upon railroads, and especially the cot struction of the great iron bridges, Conway aid Menai, were the occasion of new experiments, very numéreos and interesting. The account of thee observations, generally extended to the point oí rapture, is contained in the monograms upon tu construction of these great bridges of Stephenson, All these results have led to more exact determi nations of the specific constants relative to m They show that iron obeys the theoretic laws >. transverse flexion for slight deformations ; that IL coefficient of elasticity is reduced in very lam works; that the resistance of iron to crushinc:; generally less than the tensile resistance. The reverse is true of steel. Besides, the limit d elasticity of steel is greater for compression time for tension; while in wood and iron the limits are the same. The coefficient of elasticity for steel is not the same for compression as for extension—¿ singular anomaly, which makes rigorous calculation of flexion very difficult. MM. Collet-Meygret an J Desplaces have deduced an interesting consequence from observations made ou the Viaduc de Tarascón it is, that in steel the sections do not show everywhere an equal elasticity, so that it is necessary to distinguish the shell from the portions near the surface, giving to each ite special coefficient. The line of demarcation between the two regions seemed to be difficult to trace rigorously. Iron drawn into wire of a small diameter presents analogous phenomena; so that it has long been known that iron wires contain elements that resist extension—a property «bhxed in the oonstreebon of sospiusu.ji bridges.

Hodgkinson has made a special study of the resistance of pieces under pressure at their shuttings ; among these, upon steel columns. His empirical formulas give the breaking load of a hollow column in terms of its height, and its external and internal diameters. The load which a cobran can safely bear is fixed at a sixth of the Ъгеакшс, load given by the formula. He has shown by numerous experiments the influence of the shape of the base, whether flat or round ; and also the effect ol rende• ment. He has substituted a formula for wood instead of that given by Rondelet, and shows that the load supported by oak varies nearly as th» fourth power of the least dimension divided by the square of the length. M. Sove has substituted more convenient formulas for solid columns of steel or iron; they lead without difficulty to the choice between iron and steel for a column of given dimensions.

The observation of the vibratory motion of elastic solids leads to the determination of the co efficients of elasticity. It is the only method applicable to their plates or to wires. To this сЫ of researches belong the experiments of AVertheim. and those of M. Phillips upon the spiral réglant.

In a scientific point of view the experiments upon the resistance of materials have shown tint the hypotheses assumed in the solution of the problem of the deflexion of beams cannot be regarded as absolutely true. It is almost impossible to determine the limit of elasticity. A metallic bar, once subjected to tension, does not return to precisely its original form when the tension ceases, and its'elasticity has undergone a certain alteration. The limit of elasticity, as understood in practice, is that limit at which alteration becomes sensible by the coarse methods of experiment; but as these become more precise, it is seen that this limit gradually diminishes, and it would without doubt disappear if the processes were perfect. The limit of elasticity is certainly of great inipeirtiUice; but one should not pretend to determine it with a precision that cannot be applied in a practical problem. Late experiments have shown thei existence of certain gaps in the theory, without fiUinc them. They have shown the great complexity of the problems without suggesting the means of getting rid of it; so that the old theory of de-flexion remains the only guide that can bo confidently followed, while we wait for the perfect solution which is to come from future investigations.

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