18 Proceedings of the Royal Irish Academy. 



2la^[0-'?^■^Wa{nc,-^i)-e-^^'-')FJnc,n)\^, h- Denominator of (16)1 



(18) 

 The symbolic- term Tdidx does not occur in the minors* A^, B^, &c., nor 



in the determinants II,, and thus aflfects only the final terms of oP",, jPi, aT^s, 



eta ; it is to be interpreted in the usual sense. 



It must never be overlooked that, in <j>, the form of the numerator of the 



integrand in the double integral may be altei^ed by interchanging a and b. 



Fa and F-„ 



Art. 6. Thvs Solution satisfies the given Initial Conditions. 



Proceeding to the verification of this solution, I shall first show that it 

 satisfies the initial conditions. The initial values of 0, dipjdt are ;/<, i^ respec- 

 tively ; for, having regard to the difi'erence of notation, the expansion given 

 for <p agrees with the left-hand member of (5) as far as is necessary : that is 

 to say, the coefficient in (5) of 



in the integral which explicitly refers to the boundary i, agrees with 24X11 as 

 far as terms of index (y — 2) ; and similarly for the boundary a. 



In considering the initial value of a^t the portion whose integrand con- 

 tains uAii as a factor may be n^leeted, since this factor is one dimension 

 lower than FJjxc, ± pi], which has to be set against it in the denominator, and 

 since jIIi is of lower dimensions than F^. 



In the remaining portion, the integrand is asymptotically of the form 

 — 2€i^fC^y,dfi, so that the initial value of a^i is y^- 



In obtaining the initial value of ddt{aTi), the portion involving jiAuiJii 

 may be neglected, since it vanishes until the time (b - a)Jc. 



And the remaining portion may be differentiated with respect to t under 

 the sign of integration. For, if we do so and integrate by parts the integral 

 involving }p {u), the terms arising from the two integrals in u and from the 

 boundary terms at i are uniformly convergentf and have zero for their initial 



• -B12 denotes the analogue to ^12 at the end i. 



t For then the pulynomials in the numerator of the integrand are, for the double integrals one 

 dimension, and, for the portions of the single integrals which do not cancel, two dimensions, leas 

 than those in the denominator. 



