136 Mr. R. Carmichael on Homogeneous Functions, 



corresponding equation (b) containing any lower number of inde- 

 pendent variables. Hence is derived the following conclusion, 

 which seems to be of some importance. 



The solution of an ordinary linear differential equation of the 

 class represented by (No. 4) 



A X ' ^| + Ba/ 3 ^4 + &c = ax m + bx» + &c. . • (c) 

 dx dxP 



being given, we can at once write down the solution of a partial 



differential equation of the class represented by 



. / d*z a . d'z . \ 



A( x* ha/ -1 ?/ h&c. ) 



V d,. a clx a -^du / 



L = © m +@„ + &c. {d) 



daf dx a ~ l dy 



+ &c. 



by substituting for ax m , bx 11 , &c. the corresponding known ho- 

 mogeneous functions ©,„, ©„, &c, leaving the numerical coeffi- 

 cients introduced by the process of integration untouched, and 

 by introducing for each term in the solution of the ordinary linear 

 differential equation in which an arbitrary constant is introduced, 

 such as C m x m , a homogeneous function of the same degree, but of 

 arbitrary form in x and y. 



Thus the solution of partial differential equations of the class 

 (d) is reduced to the solution of the corresponding class (c) of 

 ordinary linear differential equations. 



8. So far we have only investigated and applied the analogue 

 of the first fundamental principle employed by Professor Boole, 



f(D).e m9 =f(in).e mG , 

 namely, 



¥{V).u m =¥(m).u m . 



By its aid we have been enabled to obtain the solutions of a very 

 extensive class of partial differential equations, and the examples 

 seem to show that the method possesses both generality and 

 flexibility. 



Let us proceed to investigate the analogue of the second fun- 

 damental principle, 



f(D).e m9 co = e mG .f(D + 7n).(o, 

 by the aid of which many results, both important and elegant, 

 are obtained in the memoir with considerable ease. 

 Puttin" x = e 9 , it becomes 



f{ v d d x) xm(0 = tm - f ( x i- { - m )- t 



