process of mean values giving an ambiguous result, such as 1, would imply that the real result is the mean between the two values of the ambiguous expression, which in the present case would be zero.

That

tan 0+tan n+tan n+tan (2 n) +........+tan(mn))÷m, where mn=2, approximates without limit to zero as n diminishes without limit, may be shown by causing n to diminish in such a manner as never to be effected by making n an odd

π 3 π
2 2

be an aliquot part of or

This may

aliquot part (as small as we please) of π.

It is then certain that the terms of the above series cancel each other, and we are not embarrassed with the

П

2

3 п
2

difficulty of proving that tan+tan =0; for we never fall on these values. Moreover, it has already appeared that the angle whose tangent is √ is x=1; which corresponds with the value derived from the expansion of tan-1 when x is made equal to -1; while the equation -1√/=1=∞ would be at variance with this expression.

tan

In conjunction with tan (X-1)=√1 it may be useful to notice,

cos (x-1)
· · √ = 1) = (e*+e ̄*); cos (∞ √ −1) = ∞ ;

1

sin (∞ √—I)=~21 (e'—e—*); sin (X√=1)=x√=1;

and to compare these expressions with the hyperbolic tangent, cosine, and sine of a real angle which are respectively 1, X, and X.

We have already remarked that the results here contemplated, whether arrived at by the principle of means, or by reference to the problems producing them, are to be regarded not as unique values, but as interpretations quoad the particular subject in hand. It is not true that 1-1+1-1+.....equals generally, or that this series has any unique value; for it may be made to represent any proper fraction by making it the limit of the series

If A and B play, and win alternately, A winning the first, the value of A's winnings is; but that is only on the assumption that there are no drawn games; for if there be m-1 drawn games after each game won by B before A wins again, the value of A's winnings is 2, being the limit of the above

n

series; and if the number of games played be indefinite, the doctrine of means leads to this identical result.

It has been suggested by Mr. De Morgan in the memoir above referred to, that the fabric of periodic series and integrals raised by Fourier, Poisson, Cauchy and others would be exposed to great danger by the production of any case in which 1-1+.... should differ from when it is the limit of a series A A+..... If this suggestion should prove to be well-founded, it would lead to great doubt as to the truth of the results obtained by these analysts; for although I am not able to adduce any instance in which the known analytical envelopment of such a series as a¤ (0) — x¤(1) +x¤(2) — xØ(3) +... differs from when x=1, yet it is easy to adduce cases in which the doctrine of mean values applied to such a series fails to produce as the limiting value. This doctrine gives as the limiting value,

__¢(0)—ø(1)+4(2)—p(3)+·······±p(n),
4(n)

n being made infinite. If p(n) be n2, so that the series becomes 1-x+x1 -x+x16 ........., this mean value is (1+5+9+13+17+.+(4p+1)) (2p+1), which, as p increases, approaches; and in this case the analytical envelopment, which can be found in the shape of a definite integral, also gives. The doctrine of means gives the same result for any other integer power of n; and probably, if the analytical envelopment were found, it would give the same value. But if (n) be of the form a", the doctrine of mean values gives as the limit of the series

x — x(α) + x(«2) — x(×3) + x(a1) — ...

and this circumstance of itself would induce me to require

to see the analytical envelopment of the series before I pronounced its limiting value to be. There are algebraic considerations which would rather tend towards the conclusion that this limit is a function of a; for when a is 1, its value is; and if a be infinite, its value appears to approximate to unity. It is perilous, however, to hazard surmises as to the value of this limit, so long as no finite equivalent for the series is produced.

Report on the Improvement of Telescope and Equatorial Mountings. By THOMAS GRUBB, M.R.I.A. &c.

THE labours of the Earl of Rosse, now only perhaps receiving due appreciation, have placed beyond doubt the practicability of producing specula for reflecting telescopes of dimensions equalling, if not exceeding, those which the conditions of our atmosphere permit of being used with advantage, combined with an accuracy of surface and consequent excellence of definition which we can scarcely either hope or desire to surpass.

Meantime the achromatic objective has received but small increment of dimension, and is now probably for ever distanced, in this respect, by its competitor the reflector. The spirited exertions of the Messrs. Chance of Birmingham have indeed produced a pair of discs suited to the formation of an object-glass of about 29 inches diameter, but these exertions have not been seconded by a corresponding spirit in Great Britain, either public or private. A few years since, the possible acquisition of an achromatic telescope, of corresponding gigantic size, was looked forward to as a national triumph, if ever accomplished; but our Government, retaining its character of proverbial supineness (if not apathy) in such matters, has allowed these splendid discs to be transmitted to a more congenial kingdom; yet even there the work seems to progress but slowly, and I apprehend that their formation into an object-glass is still a work for the future. Four years have now passed since the production of these discs, and nearly three years since, on being applied to by Messrs. Chance, I offered to form them into an object-glass. Under such circumstances it is desirable that attention should be turned to the reflecting form of telescopes as that alone suited for instruments of the largest dimensions, and important that these should receive from time to time such accessions of improvement as the progressive steps in arts or science place at our disposal.

Now the two points in which inferiority may be at present held as against the reflector are-the greater liability of the surfaces to tarnish, and the less intrinsic brilliancy of the pencil. In respect of the first, I hold the objection to be much less in amount, with good specula, than usually supposed. If we have to infer that Sir J. Herschel frequently repolished his mirrors at the Cape, we know that some specimens of optical glass have rapidly deteriorated. It was stated by Professor Moll, of Utrecht, at the former Meeting of the British Association in Dublin, that there was then in Paris an object-glass through which we might in vain attempt to look; but it is manifest that neither a low quality of speculum metal, nor glass carrying its own destruction within its substance is fitted for optical instruments, and that both should be equally avoided. As a proof of the permanence of good specula metal, I may mention that on a recent occasion, a surface twelve years polished showed an increase of only six per cent. of reflecting power on being repolished.

In respect of the second point of inferiority of the reflector, viz. the greater absorption of the incident light, and consequent lesser intrinsic brightness of its pencil, as compared with that of the achromatic. I would observe, in limine, that this difference decreases as the size of the object-glass increases, so that an object-glass of 4 feet diameter, and of a thickness adequate to resist flexure, would transmit little, if any, more light to the eye than a reflector of equal aperture as it is now possible to construct it. Such considerations do not however lessen the importance of obtaining for the reflecting telescope every possible accession as well to the permanence as to the reflective power of its surfaces, compatible with their general accuracy and perfection of figure. To the improvement of the reflecting telescope in these respects, I have lately devoted some attention; how far I have realized what is valuable remains to be shown.

So far as the Cassegrain and Gregorian forms are concerned, these improvements are based upon the employment of one or more silvered (not quicksilvered) surfaces; and my first application of it has been to that form. of the reflecting instrument which I have long preferred (not perhaps without good reason) to all others, viz. the "Cassegrain." Convinced, from previous practical working of both speculum metal and glass, that both were capable of receiving equal degrees of accuracy of surface, I conceived it unnecessary to stop to consider whether the failure of a recorded attempt to construct a reflecting telescope of quicksilvered surfaces was due to the errors of workmanship of the artist, or the formula by which he was guided, and selected the small mirror of the Cassegrain telescope for experiment.

Now the most obvious construction for a silvered mirror for such, was to form a lens (so to speak) of equal thickness throughout, having no dispersion, and therefore requiring no correction of colour, and to silver the concave surface. This construction I rejected, notwithstanding its simplicity, on considering that there would be a secondary image (coinciding nearly with the primary) formed by the outer or unsilvered surface, and producing what is called a "ghost" in the field of view.

I therefore assumed a radius of curvature for the outer surface differing considerably from that of the inner or silvered surface; and as this would produce refraction and therewith colour, it became necessary to adopt an achromatized compound of crown and flint glass. This being constructed, has proved altogether satisfactory: the inner surfaces being cemented, no appreciable loss of light occurs from using two lenses instead of one; the reflecting surface being as yet only quicksilvered, no increase of light should be expected: still, when the combination is used in a telescope, the image

appears both brighter and whiter than when using the ordinary small speculum; the image also appears perfectly free from chromatic dispersion. When the quicksilver shall have been replaced by a surface of pure silver, the increase of light will of course be equivalent to the proportionably higher reflective power of the latter, which, in the absence of good photometric observations, may be estimated at the least at a fourth. The same principles I propose to apply to the improvement of the Gregorian telescope, with inverted surfaces.

In the case of the Newtonian reflector, and where the aperture does not exceed 12 inches, the prism of total reflexion with plane surfaces, as at present occasionally used, seems hardly to admit of improvement; but for much larger apertures, and especially when we approach the size of Lord Rosse's great telescope, where the requisite size of the prism would involve the passing of the rays through about 6 inches of glass, the case is widely different, and if the difficulty, not to speak of the expense, of procuring prisms of homogeneous and perfectly annealed glass of adequate dimensions did not prevent their use, their thickness would go far to neutralize their usefulness.

The arrangement which here first presented itself, as affording some special advantages and permitting of a great reduction in the size of the reflecting prism, was to construct the prism with a converging power, and place it beyond the focus of the large speculum, so that the reflected pencil would form a secondary image to be viewed by the eye-glass instead of the primary. By adopting an aplanatic construction for the prism, the distinctness would be preserved, and the entire arrangement better (as having fewer surfaces) than the more obvious one of a small plane prism placed a short distance within the focus, and reaching (so to speak) this image with a long compound eyepiece of four lenses. Both constructions, however, include two obvious disadvantages; viz. a secondary image, illuminating the surfaces and making the field less dark than otherwise; and secondly, and which is of more consequence, a very reduced field of view.

That form of the reflecting prism which I propose for adoption in the case of large Newtonian reflectors, is as follows:-the prism is an aplanatic compound of negative or diverging power; this power is of course arbitrary or ad libitum, but I prefer that it be such as will about halve the angle of convergence of the pencil passing through it from the large speculum. Assuming this proportion to be adopted, the practical effect will be as follows:the requisite size of the prism will be just halved (linearly), the resulting image will be doubled in linear dimensions, and the magnifying power (with any given eyepiece) augmented in the same proportion. The length of the telescope will indeed be increased, but only by one-fourth of a diameter of the large speculum. This arrangement has the obvious advantages of the fewest possible surfaces, and no secondary image. It has been objected that the field of view is by it lessened. I cannot consider such to be the case in a practical sense; for even with Lord Rosse's telescope of 54 feet focus, the lowest eyepiece in general use may be doubled in all its proportions, and with such lower-power eyepiece and the proposed prism, the magnifying power and angular extent of field would correspond with these same as obtainable from the combination of the higher eyepiece and ordinary plane mirror or plane prism.

It will be observed that my proposed improvements, so far as described, relate only to substitutes for the small mirror of reflecting telescopes; and for so far I consider they may be confidently and advantageously applied. I see, however, no reason why the same may not be applied to the large specula

of the smaller reflecting instruments, and, assuming that either the Cassegrain or Gregorian form is selected, a beautiful principle of correction is indicated, viz.: let both large and small mirrors be made each of a single piece of glass, let the outer surface of the larger lens (that which when silvered becomes the larger speculum of the telescope) differ in its radius of curvature from that of the silvered surface by the least quantity which will sufficiently dissipate its reflected image in the field, and let the outer surface of the smaller lens (that which when silvered becomes the small speculum) differ from the silvered surface of the same in an opposite manner, i.c. (allowing for the distance between the two lenses) so that the colour produced by the refraction of the larger lens shall be balanced by the colour of an opposing refraction in the smaller. This done, the combination as a whole will be achromatized, and the secondary images (or "ghosts") so far dilated as to be insensible in the field.

For the great speculum of instruments of the largest class we probably must retain the speculum metal; there is, however, a construction which is possibly practical up to a considerable size, viz. that of a comparatively thin lens, silvered at the back, and supported throughout its back (or nearly so) by a thick or ribbed disc or casting of glass or metal, ground to fit with adequate accuracy.

It may be useful, in concluding this section of the subject, to make a rough comparison of the achromatic and reflector. A 15-inch reflector, in which the suggested improvements were carried out so far only as the small metal is concerned, would equal a 12-inch achromatic in light, and a reflector of 36 inches in diameter, similarly circumstanced, would be more than equivalent to an achromatic of the size of the 29-inch discs already spoken of, while, the length of the telescope being in each instance, for the achromatic, more than double that of the reflector, the expense of the mounting may be estimated as fourfold.

In passing to the second division of my subject, viz. the improvement of the equatorial mounting of large telescopes, I would first briefly advert to the several constructions in use, and which may be classed under three varieties, two of which are of English, the third of German origin. We have, then, the long-polar axis variety, which has the great disadvantage of the unsteadiness resulting from the telescope being attached to nearly the weakest part of an axis longer than itself. Secondly, we have the overhanging construction, consisting of a cone of great comparative weight and dimensions, and prolonged beyond its upper bearing in a biforked manner, thereby admitting of the telescope turning on bearings within the projecting fork. This construction requires for steadiness an unwieldy mass of moving matter in proportion to the optical power it supports, four tons being used in the case of a telescope of 8 inches aperture, a mass tenfold that required with a better construction.

The third variety, or German form of equatorial, has the advantage of the telescope being supported as close as possible to the strongest part of its polar axis; and the efficiency of such mounting is placed beyond doubt by the well known Dorpat instrument, and subsequently by the working of still larger instruments, for example, that erected many years since in this country for E. J. Cooper, Esq., where the telescope is 133 inches aperture and 24 feet focus, and which has remained in effective use, with scarcely any repair, from the time of its erection, although unprovided with a dome or other roof, a point of no slight importance when we consider the expense of such for so large an instrument, not to speak of the labour and time consumed in moving it during observation.