650 report — 1884. 



tinuous ; and furthermore, it is not shown explicitly that terms containing- 



d m 5w 



— — ", where m is greater than n— 1, do not appear at the limits. 

 dx" 1 



The following proof seems free from objection : — 

 If we adopt the notation 



d r y 



we may write 



where 



and if 



we get 



Again, observing- that 



V= dfr> 



-4 



8U = 2 jY,.8 r tf,-, 

 T -*f * 



Y d2f 



" di/dy*' 



o--U = 2JY rs S?/%-*df 



hrhyS = ^ (8yrSr_1) ~ s y +1 ^ s_1 > 



and that Y„ = Y, r , it is easily seen that any series of this kind can be reduced 

 to limiting terms + 2A r ( hy J dx. 



Now write 8y = z l d 1 y, and we get 6 2 U = limiting terms + [tB^yrydv 



after proper reductions. Since the B coefficients are functions of b, and ite differen- 

 tial coefficients, we can determine z 1 so that B = 0. Again, if 8^ be regarded as 

 constant, the whole integral vanishes, and therefore S'-U depends only on limiting 

 terms ; consequently by, or z v is a solution of the equation for by got by 



or if y= \ (x,c u c.,, . . . c. 2n ), then s, = J- will do what we want. (The strict 



J flCj 



proof of this requires that we consider AfiU in place of 8 2 U, and make A,y con- 

 stant, leaving 8 lt v arbitrary, A,y being — -). 



~i 

 We now have 8 a U = limiting terms 



tJK<tr)> :; 



Write this as = 1 2 C r | | V s. / I I dx, where in this and in what follows the 



f - 1 //WYT 



Jo ^ r I I s i ' I I d> v > where in this s 

 vitiation 1 V / / w ^ respect to x, 



dots mean differentiation I \ I I with respect to x, and make C vanish by 



choosing 8 properly. 



The value of 6 is found by considering that if 



( ) =c8, c being constant, by must be a solution of ( 1) 



and therefore 



df 



<rty' 



here s 2 = 



dv., 



