involves but one arbitrary constant, and that as a factor. We shall henceforth denote by Pi, or Pi(u), that particular form of the integral which assumes the value unity when u is put equal to unity. We shall next prove the following important proposition. If h be less than unity, and if (1 – 24h + ho) be expanded in a series proceeding by ascending powers of h, the coefficient of h' will be P, Or, (1 – 2uh + ha-1 = P,+P,h+ ... +P,h' + ... We shall prove this by shewing that, if H be written for (1 – 2uh + h*)), H will satisfy the differential equation d dH ď" 1 1 – 24h + h’; H {(1 – way ر dufth dhe (hH) = 0. = .. dH dH du =-2 H +3 (1 - ?) hHo, 1 dH And Hdh =μ-h, d dH 1 hdH .. (hH)= H +h H dh dh HTH dh H ... 2 dus dhi (NH) (hH)= ( {11 (1 -uh)} dH dh dH) d? Hence + hH - 3uHo +3 {u (1+i*)– 24°l} Ho ан) th (1H)=0. dul dh = = This may also be shewn as follows. If x, y, z be the co-ordinates of any point, ' the distance of a fixed point, situated on the axis of 2, from the origin, and R be the distance between these points, we know that, RR = x + y + (z' — 2)", 2 Now, transform these expressions to polar co-ordinates, by writing x=r sin cos , y=rsin O sind, 2 = r cos 0, and we get RP =pl-2z'r cos +2", (2 or, putting cos 0 = j, ? d du 1 H2 1 H Н or R= r z' dhe (h77). adhi (hH) + = ? h đ and dr2 R .: the above equation becomes h d? d d H 1 , du du dH = 0. du due) 4. Having established this proposition, we may proceed as follows: If p; be the coefficient of hi in the expansion of H, H1p,h H=1+ 2+ + 2*+...+ pl*+... (hH)=1.2p,h+2.3ph +. +ii+1)p;h' +... ď (hFI) or + ..h dh Also, the coefficient of k' in the expansion of dpil dus Hence equating to zero the coefficient of k', d dpil +i (i+1) p;=0. du Also Pi a = 0 P h 1 + + ...), is a rational integral function of Mo And, when j = 1, H= (1 – 2h+h)-} 1+h+ to +...+h+... (1 – 2uh +h*) * = P +Ph + ... +Pik' + ... M h h* = (P. + 1 since is less than 1, h P. P ++ + ... + minit Hence P. is also the coefficient of h-(i+1) in the expan 1 sion of (1 – 2uh + h) in ascending powers of + when h is h greater than 1. We may express this in a notation which is . strictly continuous, by saying that P,=P-(i+1)= This might have been anticipated, from the fact that the fundamental differential equation for Pi is unaltered if - (i+1) be written in place of i; for the only way in which i' appears in that equation is in the coefficient of P:, which is i (i + 1). Writing - (i + 1) in place of i, this becomes - (i+1)(-(i+1) + 1} or (i+1) i, and is therefore unaltered. P, Then -1)! 8" (2) idzi ( and let k be any quantity less than r. {x' + y +(z— k)?}}= f (z – k), and, developing by Taylor's Theorem, the coefficient of k' is 1 ď 1.2...ide 1 k k2, } 22 since z= = ur, in the expansion of which, the coefficient of k' is P. = }(1-2 P.=(-1' 1.2 ... i dz Equating these results, we get poti di i The value of P. might be calculated, either by expanding (1 – 2uk + ha) + by the Binomial Theorem, or by effecting the differentiations in the expression (–11',), gotti idzi 1.2.3 and in the result putting = M. Both these methods how ever would be somewhat laborious; we proceed therefore to investigate more convenient expressions, +1 r |