17(5 1^- FüJISAWA. 



To find X,., observe that A'., being a function of tlie differential 

 coefficients of y with respect to x only, is independent of u as a 

 function of y. Hence, putting u = y, y'r-y" successively, we obtain 



/ d"y _ ^ 



'^ = 2yX, + 2X,, 



'^ = ry'-'X.+ rir- 1)//-%+ ... + rlX,, 



\ '^ = ny^^-'X,+ n{n~l)y--'X,+ +n! X„ 



To obtain A^j, X.,,..., we may solve these equations as was done 

 by Ijertnind* ; but it is shorter to proceed as follows: — Put n = e^'-^, 

 then 



c-^"^^ = /Ai + /=^X+. ..+/%+. .. + /%,, 



and, on the other hand, 



d"c^'' _ . d"y '/? d"y^ )? d"y^ /" fZ"/y" 



~dx^ ~ ^'dr^"^ TT dx" ^ 3 1 ~dx^'^'"'^ nl dx" ' 



Multiplying the last two equations together and ecpiating the coefficients 

 of the like powers of A., w^e obtain 



* Bertraud, Traité de Calcul Différeutial et de Calcul Integral, T. I, p. 139. 



