PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 273 



which is equivalent to ^ + -- \/-i e ' ' .—=^ ^ . a;^=0 



O d j — J_) 



or zy+ Pz-^ =0, or, ifv=x^, 



6 a d xi 



V + ^- x" — - =0, which is the form integrated in Class 1. 

 6 a dx^ 



1 3 

 2. If - = 2 ~"' ^luation (2) becomes 



which is equivalent io y+ ^*^~^ g("-i)^t;D^ . ^ = 



which is of the same form as in the last case. 



2 

 B. If c is not equal to g' we have from equation (1) by (D.) 



/-D + M--J ^ 



, / — rl + cD — cjj / — D + w ^„ 1,. 



1 + aD /_D + «-^ ^ 



3- If 13^ =a this gives 



^ /-D -^ 



or i/ + b{\-cn) x" — -^ - 0, the same form as before. 



16. It would be improper to dismiss this equation without remarking the 

 fact that it would appear to have been solved by M. Besge in Liouville's Jour- 

 nal 1844, ix., 294. The solution is, however, given without any demonstration, 

 and is, if I mistake not, rather a differential equation formed than a differential 

 equation solved. The whole which appears is as follows : 



" Let in, n, p, a be functions of x, and -^ + m-^ + n — f + p v = g, the 



dx' dx d'ci 



proposed equation. 



" If we have ^ + nin—p=0, the given equation can be reduced to the follow- 



ing, ^-^ + }H!/=z, where z is obtained from the equation -^+n2=g." 



VOL. XVI. PART III. 3 Z 



