•264 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



By substituting this in equation (2) we obtain 



T - »»2a.« (^" '1-^ +^nx"-^ : + &c.) =0 



or 



d": 



die"' V2.i- 7n-y-" } di-"-''- 2. 



4 .r= rf^«-2 



1.1.3n{n-l)(n-2)d"-^z 



2.4.6 



&c. +(-1) 



n{n-l)...l 



„_: 1.1.3...(2n-3 

 2.4.6...2re 



When ~ has been determined from this equation, we shall have the complete 

 value ofy by means of Equation (1.), viz. 



y= 



d'^-'^z n d^-iz 



- + mx 



Cor. If«=3; 



d^z / 3 1 \d^z 3_d2 _S_ 



which is the same equation as that which we obtained by a totally different pro- 

 cess for determining y.^ and ?/i in Ex. 6. 



Ex. 8. 

 The solution is 



y — m X r = A 



^ dxi 



= (1 + m .c" dh (v + w) 



n d- V n d- W 



=iv + mx +w + in X j- 



d xi d xi 



where v is the same as in the last Example, and w is determined from the 

 equation 



dxi^ dxi ) 



d" - ' u 



or by writing — j^zj for w, and proceeding as in the last Example, 



„ d"v. nx"''^ rf-'-l « 



dx' 

 dn- 



A.-l« ., u / n d"n , nx 

 -m^x [x —— + s- 



.... - 1 



■) = 



+ &c. ) = X, or 



