74 PROFESSOR A. E. FORSYTE ON THE INTEGRATION 0V 



and equations of definition and derivation are 



= m + nq 

 = h + (J<1 



dv___ <h_ 

 du du 





du 



= 'J 



du 



dm , dz_ 

 du du 



dn 



dz 



du ~ du 



So far as concerns derivatives with regard to x and y, there appear to be twelve 

 equations, viz., the characteristic invariant, the three subsidiary equations deduced 



by taking 



DF 



DF 



and the eight equations of definition and derivation ; and the number of quantities 

 to be obtained from the subsidiary system is eleven, viz., a, b, c,/, g, h, I, m, 11, v, z. 

 On the other hand, F = is an integral of the system, and it is a persistent relation ; 

 consequently one of the set of three equations can be regarded as depending on the 

 other two. Again, from the equations 



~=l + np, f~ = m + 



we have 



identically satisfied in virtue of the values of dl/dy, dm/dx, dn/dx, dn/'dy, given by 

 the other equations of the set of eight ; hence one of these can be regarded as 

 dependent functionally on the rest, and the set of eight are therefore equivalent to 

 seven only. 



The whole set of equations in the subsidiary system, involving derivatives with 

 regard to x and y, thus contains tea independent equations ; and eleven quantities 

 are to be obtained. Hence it may be possible to express ten of these in terms of 

 one of them, or to express the whole eleven in terms of a single new quantity, 

 arbitrary so far as the set is concerned ; its arbitrariness will then be limited so that 

 it shall satisfy the subsidiary equations involving derivatives with regard to u. 



46. The relations thus obtained, as satisfying all the subsidiary equations, con- 

 stitute an integral of the equation. 



This result is established by an argument similar to that in 29 for the case when 

 the subsidiary system is simpler ; accordingly here it will not be repeated. 



