650 



REPORT — 1884. 



I 1 



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

 ?—- 1-, -where to is greater than ?( — 1, do not appear at the limits. 



The following proof seems free from objection : — 

 If we adopt the notation 



y 



we may write 



where 



and if 



we get 



Again, observiiip' thut 



</.t'' 



SU 



Y,. 



d'f 

 "dfdy ' 



(lit • 



and that Y„ = Yj^, it is easily seen that any series of this kind can he reduced 

 to limiting terms + SArfSy) dx. 



Now write hj^z^b^ij, and we get 8-U = limiting terms + [sB^X^iyO'f^c 



after proper reductions. Since the B coefficients are functions of s, and its diflereu- 

 tial coeilicients, we can determine c, so that Bo = (). Again, if b^y he regarded as 

 constant, the whole integral vanishes, and tlierefore fi-'U depends only on limitin;,' 

 terms ; consequently by, ur z^, is a solution of the equation for by got by 



df 



(1) 



or if 2/= (-'/C,, c, . . . Cj,,), then :i=';' ^^^^ do what we want. (The strict 

 proof of this requires that we consider AfiU in place of 8-U, and make A,?/ cou- 

 stant, leaving 8,?/ arbitrary, A,y being — "). 



We now have 6-'U = limiting terms 



.1-1 

 2 C, 



\ =, / I I d.v, where in this and in what foUowi 

 \, J j with respect to .r, and make C^, vault 



Write this as 



dots mean ditlereiitiation 

 choosing 6 properly. 



The value of 6 is found by considering that if 



\ ) = cd, c being constant, S_y must be a solution of (1) 



and therefore 



^ = (''=-V where =.. = -^ 



s the 

 ishby 



i >^ \ 



