[ 348 J 



XLVI. Illustrations of Symmetrical Integration. By the Rev. 

 Robert Carmichael, Fellow and Tutor of Trinity College, 

 Dublin*. 



] . 1 ET the partial differential equation 



"*T px* -f qy* = kz 2 



be proposed for integration. 



Writing this equation in full, we get 



X dx +y dy- kZ > 

 which is obviously equivalent to 



dz dz 



and this again to 





kz*, 



but the solution of this equation is at once 



where u is an arbitrary homogeneous function of the order zero ; 

 or 



where <j> is any arbitrary function. 



Similarly, the integral of the partial differential equation in 

 three independent variables, 



dw dw 2 dw_ k q 



X dx+ V dy +Z dz- kW > 



is 



i i 



w 3 \x y zj 



More generally, the integral of the partial differential equation 

 in n independent variables, 



m du m du , m du , m du , m 

 x m -r+y m -T-+z> m -r +w m -r- + kc.=:ku m , 

 dx dy dz dw 



* Communicated by the Author. 



