significant meaning; and be assured too that they are not uninteresting to all; to many they give the purest pleasure; and I must ask you not to grudge them that during the few papers on the higher mathematics which we shall probably have. In passing, too, I would remind you that very frequently our knowledge of natural phenomena depends on certain integrals, the properties of which can only be studied with a profound knowledge of the higher mathematics; and thus the progress of one branch of knowledge depends on another, and is frequently stopped by our ignorance of that.

I

To most of us, probably, the questions of applied mathematics will have greater interest; we are more familiar with the laws of nature, the mathematical interpretation of which, mixed mathematics, as they are called, take cognizance of; we most eagerly catch at the results of those laws. Consider the Newtonian law of gravitation in its most general form; in its highest development in the lunar and planetary theories, a dry mathematical paper will thin our room; an astronomical paper will often fill it; and now too, perhaps, more than heretofore; for our interest in the subject has been keenly aroused of late. The lunar disturbances have been, as you know, calculated with greater precision, and new results have been arrived at, which exhibit certain discrepancies relatively to the old. need do no more than allude to what has lately taken place at our own Royal Astronomical Society and at the French Academy; and express a hope that we shall have some communication on this subject from those who are here present, and are so well qualified to give it. Mathematicians, however, have been startled by an announcement that "what is commonly called mathematical evidence is not so certain as many persons imagine; and that it ultimately depends on moral evidence;" and moreover we are told that the "results of long and complicated mathematical calculations are not more than probably true." This we can hardly believe; it takes us quite by surprise, and we hope for further light; if, however, we must wait for light, we must wait patiently; let us not forestal a conclusion which many of us venture to think is as yet, not to say more, unproved; let us wait for the new lunar theories, which are as yet unpublished, and for the new lunar tables, which are the results of these theories. I am told, however, already that Baron Plana has corrected his calculations, and that he finds the results arrived at by Delaunay and Adams to be in accordance with his amended formula. These new lunar calculations have taken us by surprise; but again I would say let us wait, "magna est veritas et prævalebit."

We are desirous, so far as is possible consistently with the convenience of contributors, to take the papers on mathematical subjects on the early days of our meeting; and we shall be glad therefore if members who have papers on these subjects will announce them to the Secretaries without delay. And before I proceed further, we have a debt to pay, due by the cultivators of these branches of science, to those who have lately contributed reports on particular parts of our science to the British Association;-to Mr. Cayley for his report on the present state of Theoretical Dynamics, and to Mr. Smith for the first part of his report on the Theory of Numbers. It is only they who have had to go through the existing literature in any one problem, say the Lagrangian equations, or the theory of the motion of a material system, that can form an adequate value of such papers as those I refer to the literature is catalogued, indexed, and analysed; we know thereby all that has been done up to a certain point, and in our subsequent investigations our commencement starts from the close of other men's labours. We are hereby prevented from travelling over other men's ground; and we avoid that most unsatisfactory plagiarism of them, "qui nostra ante nos dixerunt." Vast and various are the benefits of our Association; but I am inclined to consider as one of the greatest, the series of valuable reports which our published volumes contain ; and those last reports to which I have referred, for their learning, their deep research, their comprehensive views of the theories explained in them, will maintain the character shared by their predecessors. While we lament the loss of Dr. Peacock and others, to whom we owe the very able reports contained in the early volumes of our proceedings, we are proud to have worthy successors in our present talented contributors.

We propose, next in order, to take those papers which treat of subjects within the grasp of mathematical symbols, at least partially, if not wholly; those whose laws

are sufficiently general for functional symbols, and from particular forms of which by mathematical processes other truths may be derived. Such are the subjects of Light, Heat, Sound, Electricity, Magnetism; we propose to take these subjects in the latter days of this week, and the first day of next. We shall, of course, consult the convenience of contributors; but it will tend, we think, to the orderly arrangement of our business if this order can be adopted. Vast indeed in their subjects are these sciences; and as discoveries are being daily made in them, we have a right to expect some interesting communications, either in the way of mathematical deduction from received laws, or as mathematical explanations of observed phenomena, or as simple experiments. I cannot help observing here the advantage of combining these sciences in the same Section with pure mathematics; it seems to indicate that the laws of all are to be brought to the same test, to the never-failing, to the unerring accuracy of measurement and number; we show hereby the character of the knowledge we are in search of; not fortuitous observation, but precise laws. The mind will wander in its imagination; there is, indeed, no boundary to it; once, however, bring it back to the severe test of number and weight and measurement, and the discovery or the observation becomes valuable for its precision; it thus leads to general laws, and sound mathematical reasoning derives from them the results they are pregnant with.

And, finally, we come to the facts of meteorology and its kindred subjects, many of which are scarcely yet brought within any law at all; analogies have been traced, and concurrent events have been indicated in many cases; little, however, has been done towards a satisfactory proof of a connexion between cause and effect. It is true that curves are traced, which purport to exhibit these effects; and they do so most graphically; but, as mathematicians say, these curves are traced only by points, and the law is not known, or, in other words, we do not know the equation of the curve; so long as this is the case, our knowledge lacks precision. These papers, however, are frequently valuable, because they supply us with accurately observed facts, which will doubtless hereafter be brought within a law. This, however, I suppose at present to be the state of the case; but we must not despise the lesser light because we have not the greater. I cannot pass over this class of papers (papers of observed facts) without alluding to the loss which we all feel in the death of the late able Professor of Geometry, Professor Baden Powell. For some years past has he continued his reports on the meteors or falling stars, or whatever you call them; this year we have his last report, which, indeed, he has not lived to finish, but has been placed in the hands of Mr. Glaisher, and completed by him. In some of these subjects we shall, I hope, obtain large accessions to our knowledge. Some few years ago I remember reading a complaint made by an eminent philosopher on the decay of mathematical knowledge in Great Britain, and especially in that of physico-mathematical knowledge. It is not my duty to make invidious distinctions; but I am sure I am repeating the now common opinion of foreigners when I tell you that that complaint was made in quite the infancy of some of our older philosophers, and before the days of Cayley, Sylvester, Boole, Maccullagh, Stokes, W. Thomson, and Adams. To this revival of science amongst us, doubtless, many causes have contributed; and I believe that the periodical meetings of this Association have done good service towards that revival; we have hereby become acquainted with others who are engaged in the same pursuits as ourselves, and stores of knowledge are communicated. Let us, however, bear in mind that our Association is formed for the advancement of science, and that we do not meet to hear of old things again in the old form; our motto is "progress." Old things we do not discard, for they may be put before us in new forms: but we meet especially to promote the advance of the boundaries of natural knowledge, and we ask our members and others to lay before us the results of their investigations. And not only in the papers which shall be read, but also in the elucidation of any difficulties which authors may favour us with, and in the discussions which it is my duty to invite you to take upon these papers, will additions to our knowledge be made; and many remarks will, I venture to think, be made pregnant with matter for thoughtful meditation hereafter. In all these discussions difference of opinion will doubtless arise; but I am sure that a spirit of friendly and mutual concession will prevail; and that in our search after truth we shall gladly and readily attribute to those who differ from us the same pure motives which we claim to ourselves.

On some Solutions of the Problem of Tactions of Apollonius of Perga by means of Modern Geometry. By Dr. BRENNECKE, of Posen.

The author suggested a new solution, depending on a remarkable property of the centres of similitude of three given circles; e. g. a circle described around an external centre of similitude, with a radius equal to the geometrical proportional of its potential distances from the two circles, intersects all homogeneously touching circles orthogonally (around an internal centre all heterogeneously touching circles). Such a circle is called a potential circle. To get the two circles which touch the three given circles simultaneously internally or externally, take two external centres of similitude, draw the two potential circles, find their radical axis, which will contain the centre of similitude of the two circles which cut the three given circles in the same time externally or internally. By combining the three external centres of similitude, you find three potential circles and three radical axes, which all three coincide. Having found this straight line, which contains the centres, it is easy to find the centres themselves by introducing a fourth circle, the reflected mirror-image as it were of any of the three given circles, by means of the found radical axis, and finding out the two circles which touch the two symmetrical circles and any one of the three given circles. Dr. Brennecke has treated the subject at large in a book which has just now been published at Berlin, 'Die Berührungsaufgabe für Kreis und Kugel,' Th. Chr. Fr. Enslin, 1860, 8vo, illustrated by eighty-four diagrams, in which all information will be found concerning the most renowned problem of geometry, concerning the problem of tactions of three given circles or four given spheres.

On a New General Method for establishing the Theory of Conic Sections. By the Rev. JAMES BOOTH, LL.D., F.R.S.

On the Relations between Hyperconic Sections and Elliptic Integrals. By the Rev. JAMES BOOTH, LL.D., F.R.Š.

In this communication the author extended the analogies that the Continental and English geometers had established between elliptic integrals of the third order under the circular form, and the arcs of spherical conic sections, to the corresponding relations between elliptic integrals of the third order and logarithmic form to the arcs of curves described in the surface of a paraboloid.

On Curves of the Fourth Order having Three Double Points.

By A. CAYLEY, F.R.S.

The paper is a short notice only of researches which the author is engaged in with reference to curves of the fourth order having three double points. A curve of the kind in question is derived from a conic by the well-known transformation of substituting for the original trilinear coordinates their reciprocals; and the species of the curve of the fourth order depends on the position of the conic with respect to the fundamental triangle.

On the Trisection of an Angle. By PATRICK CODY.

On the Roots of Substitutions. By the Rev. T. P. KIRKMAN, A.M., F.R.S. To determine the number of roots of a given degree, of a substitution & made with n letters, and of the rth order. A substitution which has not two circular factors of the same order, has no roots which are not found among the series

of its powers.

1+0+0+..+ør-1

A substitution which has two or more circular factors of the same order, will have roots of an order superior to its own, and therefore not among its power. Thus the substitution of the 3rd order made with 9 elements,

has 1 square root of the 3rd order, 9 square roots of the 6th order, 9 fourth roots of the 6th order, 18 cubic roots of the 9th order, and 18 sixth roots of the 9th

order. These roots can be enumerated by a simple general method for of any order, made with n letters.

The fundamental theorem is the following:

If n=Aa+Bb+Cc+......, the number of different groups of the order K, which is the least common multiple of ABC...

...., of the form 1, 0, 0, ... OK−1,

where has a circular factors of the order A, b of the order B, &c., is (πn=1.2. 3

RK being the number of integers, unity included, which are less than K and prime to it.

of the 6th order. Every group (H) contains a group (G), namely,

Ιφ' φ',

and of the 6th order is the square root of 2 of the 3rd order, and the fourth root of o of the 3rd order. Also ' of the 6th order is the square root of 1, and the fourth root of p2.

The number of groups (H) being nine times that of the group (G), the group 1002 will be comprised in nine different groups (H); that is, Ø has nine square roots of the 6th order, and nine fourth roots of the same order.

of the same order. There are six times as many groups (J) as groups (G). Therefore

108

will be found in six groups (J), and either or 2 has 18 cube roots, and 18 sixth roots all of the 9th order.

In the same manner it is easily proved that the substitution of the 2nd order(n=8),

0'-34127856
12345678'

which has four circular factors of the 2nd order, has twelve square roots all of the 4th order. These form with unity and ' the two groups following,

which are of the form (IV.) discovered by Mr. Cayley (Phil. Mag. vol. viii. 1859, p. 34), who there first enumerated the forms of groups of eight.

Two such groups can be completed with unity, and any one of the

substitutions of the form '.

π8 R.4.2

=7.6.5.3.2

It is easy to form groups of Mr. Cayley's form (II.); e. g.,

12345678

34127856

23416785

41238567

56781234

78563412

67852341

85674123

which is one of the grouped groups whose general theory I have handled in a memoir which will shortly see the light.

On a new Proof of Pascal's Theorem. By the Rev. T. RENNISON, M.A.

On Systems of Indeterminate Linear Equations.

By H. J. STEPHEN SMITH, M.A., Fellow of Balliol College, Oxford. The object of this communication was to point out the connexion which exists between particular solutions of indeterminate linear equations, and their most general solution. The principle upon which this connexion depends may be explained in a very particular case. Let the sytem of indeterminate equations reduce itself to the single equation

Ax+By+Cz=0,

... (1) in which we may suppose A, B, C to have no common divisor; let also a, b, c and a', b', c' be two different solutions of that equation in integral numbers; then, if the three numbers

bc'-b'c, ca'-ac', ab'a'b

(2) admit of no common divisor, the complete solution of the indeterminate equation is contained in the formulæ

in which t and u are absolutely indeterminate integral numbers; but if the condition (2) be not satisfied, the formulæ (3) will not represent all, but only some of the solutions of the equation (1). If, therefore, by any method, as for example that of Euler, we have arrived at formulæ of the type of the formulæ (3), which demonstrably contain the complete solution of the indeterminate equation, we may be certain that the three numbers analogous to the numbers (2) admit of no common divisor. Thus, by applying Euler's method of solution, which is explained in most books of algebra, to the indeterminate equation Ax+By+Cz=0, we obtain the solution of a celebrated problem, first considered by Gauss in the 'Disquisitiones Arithmetica,' of which the following is the enunciation.

"Given 3 numbers A, B, C, to find six others,

such that

a, b, c,
a', b', c',

A=bc'-b'c, B=ca'-ac', C=ab'—a'b.”

Other methods more symmetrical, and perhaps not more tedious than that of Euler, were also suggested in this paper for the treatment of indeterminate equations, and for the resolution of an important class of arithmetical problems which depend on those equations in the manner just explained.