138 Sir. R. Carmichael on Homogeneous Functions, 



F(V).0 m W=0 m JF(V)W+pF(V)W+ ^.F"(V)W + &c.| 



F(V).W0 m =0 m |F(»O.W+^ ) .VW + ^ ) .V 2 W + &c. J \ 



which are in reality but the evolutions of 



© m .F(V + m)W, 

 according to ascending powers of m and V respectively, may 

 possibly not have occurred to the reader, and are brought pro- 

 minently forward by the general theorem. 



An adequate application to the subject of partial differential 

 eqxiations, of a principle corresponding to that which Mr. Har- 

 greave has employed with so much success in connexion with the 

 subject of ordinary differential equations, would extend the pre- 

 sent paper much beyond due limits, and offers too many difficul- 

 ties to be treated of within a confined space. It will be suffi- 

 cient, then, to endeavour to evolve the various consequences of 

 the more simple theorem given in the preceding number. 



10. By the aid of the two fundamental principles mentioned, 

 Professor Boole has shown that ordinary differential equations of 

 the form 



(a + bx + ex* + &c.) %^ + (a' + b'x + c'^ 2 + &c.) =-= + &c =X 



may always be reduced to the form 



(}) (D).u + (f> l (D).eOu + ( j) 2 (J)).e^u + &c. =0. 



It is obvious that the solutions of such ordinary differential 

 equations may, then, be exhibited in the shape 



r{^ o (D)+ e y.c/) 1 (D + l)+ e 20.0 2 (D + 2) + &c.}- 1 .0 

 u=l + 



L{0 o (D)+ e e. < /> 1 (D + l)+ e 29.^(D + 2) + &c.}- 1 .O. 



Similarly, the solutions of partial differential equations of the 

 form 



(0 o + 1 + &c.).«I>(V)e + (H o +H 1 +&c.).^(V)^ + &c. = n 

 can be exhibited in the shape 



r{(0 o +© 1 +&c.).O(V) + (H o +H l +&c.).^(V)+&c.}-'.n 



+ 

 l{(0 o +0 1 +&c.).$(V) + (H o +H 1 +&c.).¥(V)+&c.} -0. 

 The following examples will illustrate this result. 

 (I.) xp + yq-{® m +®n)z=0, 



0m ©n 



