where Po, P1, P... are Laplace's, or in this case more properly Legendre's, coefficients*. Hence by expanding and differentiating with respect to c, we have

We are not of course concerned with the constant term in the latter of these two expressions. For the normal velocity (v) at the surface of the sphere we get by differentiating with respect to r, and then putting r = a

First suppose the point P outside the sphere, let the sphere be thought of as a solid sphere, and consider the motion "reflected" (p. 28) from it. The reflected motion being symmetrical about the axis, we must have for it

where Qo, Q1, Q2...., are Laplace's functions involving only. This gives for the normal velocity (v) in the reflected motion at the surface of the sphere

* The functions which in Art. 9 of the paper "On some Cases of Fluid Motion" (p. 38) I called "Laplace's coefficients," following the nomenclature of Pratt's Mechanical Philosophy, are more properly called "Laplace's functions;" the term Laplace's coefficients" being used to mean the coefficients in the expansion of [1 − 2e {cos @ cos 0' + sin ◊ sin 0' cos (w – w') } + e2] ̃1⁄2,

Ө

to be understood according to the usual notation and not as in the text.

This is identical with what (1) becomes on writing m', c' for m, c provided that

Hence the reflected motion is perfectly represented by supposing the sphere's place occupied by fluid within which, at the point P' in the line OP determined by OP' =c', there exists a singular point of the same character as P, but of opposite sign, and of intensity less in the ratio of a3 to c3.

The case of a spherical mass of fluid within a rigid enclosure and containing a singular point of the second order with its axis in a radial direction might be treated in a manner precisely similar, by supposing the space exterior to the sphere filled with fluid, taking to represent the reflected motion in this case, instead of (5), the corresponding expression according to ascending powers of r, and comparing the resulting normal velocity at the surface of the sphere with (3) instead of (4). This is however unnecessary, since we see that the relation between the two singular points P, P' is reciprocal, so that either may be regarded as the image of the other.

Suppose now that we have two solid spheres, S, S', exterior to each other, immersed in a fluid. Suppose that S' is at rest, and that S moves in the direction of the line joining the centres, the fluid being at rest except as depends on the motion of S. The motion of the fluid may be determined by the method of successive reflections (p. 28), which in this case becomes greatly simplified in consequence of the existence of a perfect image representing each reflected motion, so that the process is identical with that of Thomson's method of images, except that the decrease of intensity of the successive images takes place according to the cubes of the ratios of the successive quantities such as a, c, instead of the first powers.

If a sphere move inside a spherical envelope, in the direction of the line joining the centres, the space between being filled with fluid which is otherwise at rest, the motion may be determined in a precisely similar manner.

If two spheres outside each other, or just touching, be connected by an infinitely thin rod, and move in a fluid in the direction of the line joining their centres, we have only to find the motion

due to the motion of each sphere supposing the other at rest, and to superpose the results.

I should probably not have thought of applying the method to the solid bounded by the outer portions of two intersecting spheres, had not Professor Thomson shewn me that it was not limited to the cases in which each sphere is complete; and that although it fails from non-convergence when the spheres intersect, yet when the exterior angle of intersection is a submultiple of two right angles the places of the successive images recur in a cycle, and a solution of the problem may be obtained in finite terms by placing singular points of the second order at the places of the images in a complete cycle.

The simplest case is that in which the spheres are generated by the revolution round their common axis of two circles which intersect at right angles. In this case if S, S' are the spheres, O, O' their centres, 0, the middle point of the common chord of the circles, the image of O in S' will be at 01, and the image of 0, in S will be at O'.

1

1

1

2

Let a, b be the radii of the spheres; c the distance (a2 + b2) of their centres; e, f the distances a2/c, b/c of 0, from 0, Ο' ; C the velocity of the spheres; r, the polar co-ordinates of any point measured from 0; r,, 0, the co-ordinates measured from 01; r', 'the co-ordinates measured from O'; 0, 01, 0' being all measured from the line 00'. If S' were away, we should have for the fluid exterior to S

1

For the image of this in S' we have a singular point at 0, for which

and for the image of this again in S we have a singular point at O' for which

which is precisely what is required to give the right normal velocity at the surface of S. Moreover all the singular points lie inside the space bounded by the exterior portions of the inter

secting spheres. Hence the three motions together satisfy all the conditions of the problem, so that for the complete solution we have

Just as in the case of a sphere, if a force act on the solid in the direction of its axis, causing a change in the velocity C, the only part of the expression for the resistance of the fluid which will have a resultant will be that depending upon d C/dt. This follows at once, as at pp. 50, 51, from the consideration that when there is no change of C the vis viva is constant, and therefore the resultant pressure is nil. If we denote by M'd C/dt the resultant pressure acting backwards, we get for the part of M' due to the pressure of the fluid on the exposed portion of the surface of S',

taken between proper limits. Putting b cos' = x, we have

Expressing cos, cos 01, cose' in terms of x and r, x and r1, x, and changing the independent variable, first to x, and then in the first term to r and in the second to r1, we have for the indefinite integral with sign changed

1

which is to be taken between the limits r =ator=c+b, r1 = ab/c to f+b, x=-ƒ to b. The part of M' due to the integral over the exposed part of the surface of S will be got from the above by interchanging; and on adding the two expressions together, and putting ƒ= b2/c, c = √(a2 + b2), we get for the final result

2

M' = "p3 {4c3 (a3 + b3) — 2ao — 3a1b3 — 6a3b3 — 3a2b1 — 2bo}.

πρ
3cs

3

When one of the radii, as b, vanishes, we get M' =πрa3 as πρα it ought to be.

[From the Transactions of the Cambridge Philosophical Society,
Vol. VIII. p. 533.]

ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES.

[Read December 6, 1847.]

THERE are a great many problems in Heat, Electricity, Fluid Motion, &c., the solution of which is effected by developing an arbitrary function, either in a series or in an integral, by means of functions of known form. The first example of the systematic employment of this method is to be found in Fourier's Theory of Heat. The theory of such developements has since become an important branch of pure mathematics.

Among the various series by which an arbitrary function f(x) can be expressed within certain limits, as 0 and a, of the variable æ, may particularly be mentioned the series which proceeds according to sines of πx/α and its multiples, and that which proceeds according to cosines of the same angles. It has been rigorously demonstrated that an arbitrary, but finite function of x, f(x), may be expanded in either of these series. The function is not restricted to be continuous in the interval, that is to say, it may pass abruptly from one finite value to another; nor is either the function or its derivative restricted to vanish at the limits 0 and a. Although however the possibility of the expansion of an arbitrary function in a series of sines, for instance, when the function does. not vanish at the limits 0 and a, cannot but have been contemplated, the utility of this form of expansion has hitherto, so far as I am aware, been considered to depend on the actual evanescence of the function at those limits. In fact, if the conditions of the problem require that ƒ (0) and ƒ (a) be equal to zero, it has been