lU 



Proceedings of the Royal Irish Academy. 



investigated, ha^dng the values of I, X assigned. If Si, S^, be functions corre- 

 sponding to two different possible values pi, jh of j^, from the equations 



vd-Silchf = {v\- + ih + il^y) >S'i, 

 vd'SJdj/ = (vX' + jh + iliiy) So, 

 v{S,d'S,ldf - Sd'SJdif) = (2h-2h)SiS„ 

 and by integration between the limits ± a, 



there results 



(ih-pi) 



SiSidif = V 



S^dS^ldij - SdS^ldy 



(62) 



If pi and pi are different values for which Si, So vanish at the limits, tliis 

 gives 



" SiS4y = 0. (63j 



If, in the formula (62), we write jh = 2h -'<- ^pu divide by Spi, and then 

 suppose d2h to diminish indefinitely, we obtain 



Si^dy = V 



'"' dydpi dy dpi 

 dSi dSA^ 



(64) 



dy dpi\_a 

 since Si vanishes at both limits. 



Thus, if we assume the possibility of expanding an arbitrary function, /(^) 

 in a series of the form 



^A,.S,.{y), 



the coefficients are from (63), (6-4:) determined by equations of the form 



dSr dS.A'' r 



vAr 



f{y)Sr{y)dy. 



(65) 



dy dpr Ya 



Should the period-equation have a double root g, in which case that 

 portion of the complete disturbance whicb involves c^' takes the form 



AS^^ + Bif'dSjdp + tSeP'), 



the expansion of f{y), the value of S at the time t = 0, has to include a term 

 BdS\d.p as well as AS, and (65) fails to determine A, B. The investigation 

 necessary to find their values is somewhat longer, and it appears unnecessary 

 to give it. 



I have not succeeded in applying these formulae to au}' initial distiu'bance 

 of the simplest type, such as that discussed by Lord Kelvin. Towards so 



