DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



49 



28. We consequently begin with the generalisation of AMPERE'S method. Let the 

 variables be changed from x, y, z, to x, y, u, where u is a function of x, y, z, as yet 

 undetermined, so that 2 is a function of x, y, u, as yet also undetermined. With the 

 notation previously adopted, we have 



and therefore, from the equations involving derivatives with regard to x and y alone, 

 it follows that 



dl dn 



a = j, - P T,: + l rc > 



dx 



ihn dn 



b- q ,- + q-c 



dy HI/ 



dn 



dm 



- 



dx 



dl 



so that we have 



dn 



7- + P f l c 



dy 



dn 



\ 



dm 



du (U 



du 



which is the condition in order that the necessary relation 



dij \ do; / ~~ dx \ dy 

 be satisfied. 



When the postulated equation of the second order is such that, on the substitution 

 of the foregoing values for a, b,f, y, h, it has a linear form in c, let it be 



J + cT = 0. 



Suppose that the variable u (or z as a function of x, y, u) is determined so that 

 - . 1 = 0, 



which, after the earlier explanations, is the characteristic invariant ; then we 

 have also 



J= 0. 



VOL. CXCI. A. H 



