DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



Application to V z v = 0. 



47. The investigations as to the potential equation, given in 31, did not require 

 the consideration of derivatives of , b, c,f, g, h ; but the example can be used to 

 indicate the method of proceeding when such derivatives must be taken into account, 

 as in the preceding theory. 



For the equation 



a -f b + c = 0, 



the special subsidiary equations involving derivatives of a, b, c, with regard to 



x and y, are 



da dn dh da 



-; p v + -r- - q T~ = 



dx d.i' dy dy 



dh df db df 

 p -i- _j_ q --- = 



dx flu di/ dy 



dq df df 



-Z- in -|- ~^ 



d.r r dx di/ 



dc 



tlie characteristic equation is 

 and the relations of identity are 



p* + <f +1 = 0; 



d<l <l'i d(/ 



<lh 



<i(L 



(h, 



df db 



df 



df 



J 



But the latter can be replaced by the equations of definition, viz., 



dl 



dl 



together with 



it is, indeed, from the first two columns of these equations of definition that the 

 relations of identity are deduced. 



L 2 



