132 Proceedings of the Eoyal Irish Academy. 



The values of -^ then proceed simply in powers of /ja*, all other terms being 

 omitted. Equation (68) given by the boundary-conditions becomes 



, ^2kW 153601^V 1426* yt-W 



The lowest root of this is approximately 



Cpla^lix = - ika,' = 107. (63) 



On the other hand, the odd forms of \p lead to the equation 



^ JcW 31815%, ^, 625%^ ,, . .„.. 



^ ^ 69300 " -Lp- ^^^^ ^ 10^ ^ "^ ^ • • • = ^' (^^) 



and the lowest root gives approximately 



Cplay^ = - ika,' = 265. (65) 



Considering then the even forms of ip, one of the series whose lowest term 

 is unity is 



i.ot = l + ^+{3 + k)^+{4 + 12k)^ + {5 + 50k + 9k^)^+{6 + UOk+17m^ 



+ (7 + 315^'+ 1189F + 9.17^:0 4^ + (8 + 616^^+ 5144^»2 + 3960P) ,-^ 

 ^ ^ I 12 ^ ■^ I 14 



+ (9 + 1092/J+ 16974Z;'- + 37492^;^ + 9.l7.25^^0 -^ + (. . . 122490A;*) t^ + . . . 



Li5 LI? 



(66) 

 and that whose lowest term is a^/2 is 



^3 = j^ + |^ + (3 + 5A-)j^+(44-28A.)i^ + (5 + 90A; + 5.13Z;'-)j^ 



+ (6 + 220Z^ + 606/v-) ,-^ + (7 + 455Z: + 30S7k' + 5.13. 21^-=') -^ 

 ^ j 12 ^ ^ I 14 



+ (8 + 840/j + 10968/r + 17880A;3) "'' 



L 



16 



+ (9 + 1428^'+ 32094/j- - 122468P + 5.13.21.29A;0 pU + . . . 



18 



+ (. . . 669210/.^) j^ + . • . (67) 



The boundary-conditions u = 0, v = evidently give 



^P.dxp^/da - ^p.d-4jjda = 0, (68) 



* These numbers are only approximately correct. 



t The boundary- value of a is denoted by ai. 



\ Probably the numerical work would have been simpler had I chosen ;fo - ^2, instead of ifo- 



