6G 



PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



on the supposition that u is the third variable : that is, in these partial differentia- 

 tions in the subsidiary system, u does not vary. Consequently J = is expressible 

 in the form 



JL(J _ n p) + ^(m- nq) + K*V = ; 



and therefore 



(1 np} + i r ( m ~ nc l) = f 2 - = K- (I + np} 

 ilc~ v dx dy v ax 



dx dy 



so that 



d~ dr 



- (I - np) + -- (m - nq) = - K S ^ = - /c 2 (m -{- nq) I 



ft 



K~ m = i r "r <7 jl !?* 



/ V <fa r^/y J </y- 



Operate on the first equation with d*/dy z + K~, on the second with d z jdx dy, and 

 subtract ; then, dividing by K", we have 



f- 



~ PK ' 



Similarly, operating on the first equation with d*/dx dy, on the second with 

 d*/dx* -\- K 1 , subtracting, and dividing by K 2 , we have 



+ I o + * 2 ) m = I <7 -':' . + 1p - - + '77-, <7 



- dip I \ 2 (/.';- A das dy * dy~ i 



It therefore follows that some function w of x, y, and u exists such that 



d-w d-w d-w 



= P -pr + 2? p 



X (/,!!- J (/,'.' rt?/ - 1 (?W 3 



dho , d-w 



m = q - -f 2p j-j- + 5 ^/c 2 w i. ; 



* rfa^ * r/A'rf?/ J r7w 3 * 



but, so far as the subsidiary system is concerned, w is perfectly arbitrary a result in 

 accordance with the general explanation of 30, there being only four functionally 

 independent equations for the preliminary determination of five quantities. Also 



_ 



m + nq = 2 (p -r-r- -f q - = 

 \ dxdy * dif 



do 

 - 



dv 





 dy 



