48 



PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



In the first section ( 2), a generalisation of AMPERE'S method was dealt with very 

 briefly, partly because that method and DARBOUX'S method apply most effectively 

 to equations in which (with few exceptions) the derivatives of the highest order 

 occur linearly ; and, of the two, it is DARBOUX'S method which can be more effectively 

 applied to other equations. The fact that the characteristic invariant was resoluble 

 proved of material importance in the general theory. 



It is to be remarked, however, that some of the equations which occur most 

 frequently in mathematical physics, for example 



8e 



3,6 i3 8>/ 3 c;'~ ot 



8-V c-r 8V 8-V 



-\ ~~; ~r~ ^ , +" ^ T C* ~ 



8.-; 3 ^ ?/ tV ct- ' 



the latter two being, for purposes of application, made to depend upon the equation 



32,. ^i,. PJ,. 



"^ O I ^ ) |~ ^ o ~ ~~ ~~* K V 



<.:,* d/ - 



belong to the class which have their characteristic invariant not resoluble, and at the 

 same time are linear in the derivatives of the highest orders that occur'. Accordingly 

 both AMPERE'S method and DARBOUX'S method generalised can be applied to such 



equations. 



Moreover, the generalisation of AMPERE'S method can also be applied to equations 

 of the form 



of 



wher 



e = a, h, y, 

 ft, (>, f, 

 1 ff, f, c, 



and the quantities 6, A, . . ., H, A J; . . ., Hj, U, do not involve derivatives of the 

 second order. For, when the equation is transformed by the relations of 1, it takes 

 the form 



J + cl = 0, 



where I = is the characteristic equation ; in other words, taking account of 1 = 0, 

 we must associate J = with it as an equivalent to the postulated equation. 



