134 Mr. R. Carmichael on Homogeneous Functions, 



6. As an example of this method of solution of partial dif- 

 ferential equations, let it be required to find the integral of 



x i r + 2xys + y' i t— n(xp + yq — z)=0. 

 When thrown into the symbolic shape, this equation becomes 



V(V-l)z-n{V-l)z=0, 



and the solution is given by 



1 N N' 



(V— ft)(V-l) V— » V — 1 



or is at once, including N and N', in the homogeneous functions, 

 which are given in degree but arbitrary in form, 

 z = u n + u 1 *. 

 As a second example, required the integral of 

 x*r + 2xys + y*t = ® m + ® „, 

 where ©„,, ©„ are given homogeneous functions in x and y of 

 the mth and nth degrees, respectively. Then 



{©,„+©„}+* ,:.(>, 



" V(V-l) l ■ ' V(V-l) 

 or, by (1), 



Z ~ m{m-l) + nJn~-[) + ^>+*f 



which is the required solution. 



As a third example, let the integral of the partial differential 

 equation of the third order in three independent variables x, y, z, 



3 d 3 u „d 3 u 3 d 3 u ~] 



X lx^ +r lf +Z 1? 



> =<£> 4-cb 



3 



* If n = t-, this value of z renders the integral 



m — 1 



ff{px+qy—z)mdxdy, 

 a maximum or a minimum within certain assigned limits ( Jellett's Calculus 

 of Variations, p. 253). 



In general, by the method stated above, it can be readily seen that the 

 form of the function w, which, for certain assigned limits, renders the sym- 

 metrical multiple integral containing p independent variables 



f, r, r, / dw d\D dw „ \m 



fdxfdyfdz.\x Tx+VTy +z-^ + Skc.-w) , 

 a maximum or a minimum is, as before, 



W=Un+Ui, 



where 



,. J»+l. 



w — 1 



