40 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



here, the term in r & is made to disappear from each of the equations, and thus 



we have 



Ap 2 + Hpq + B</ - Gp - F<7 + C = 0, 



the characteristic invariant. (We hence notice that this is a particular illustration of 

 the remark in 16, viz., if two equations are compatible with one another, their 

 characteristic invariants are either the same or, being resoluble, have at least one 

 factor common.) 



The six equations, after some reductions of an easy character, take the form 



- 



dx dx / \ dx dy i \dy * dy / dx Ay 



dy j dx dy 



For the aggregate of differential equations in the system, we have one pair 



do dv 



- = I + np , ~ 



do dv 



= I + np , ~ = m + nq ; 



three pairs, one of which is 



dl dl 



te= a + W> d ,,= h + 9P-> 

 six pairs, one of which is 



du da 



^== + ip, ^ = ^o + i?; 



one characteristic equation, 



A = 0; 

 and the above six equations. 



The three equations of 18 become three integrable combinations of the above six 

 with the foregoing equations in the six pairs. 

 The number of quantities to be determined is 



10 quantities . . . . ..... S ; 



6 ,, .... a, ...,/*; 



. . . . /, m, n ; 



1 quantity . . . . v ; 



1 . . z; 



