56 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



Substituting these values, we have 



d*w 



d-w 



M= PjZ+^tedj-Pdf 



2m = 



2n = 



dii? 



d-iv 

 ~dx 2 



2 P^. 



d-w' 



dy* 



dho 



dx dy 



dy\ 



Thus far account of variations with regard to x and to y has been taken. But, 

 as regards variations of u, we have 



dz 



that is, 



d I dw 



dv 



-j- = n 

 du a tt, 



dw 

 dy 



-w d-w 



where A denotes 1 + xp' + yq. This is the equation of limitation upon the form 

 of iv ; if its general integral were known, the general value of v could be deduced. 



2. Before proceeding with the consideration of this result, it is worth noting the 

 relation of the equations 



; (>S M 



l-np= J-, m - nq = - 



dx ' 



to the original equation. Because 



it follows that 



where A denotes 

 so that 



= u + xp 



1. Sit, 1 (hi- 

 p dx q 3// 



z 



1 + xp + yq, 



r\fc 



aT 



dy 



?f 



m-nq= - -p 



G/v 



- np = aT 



dy 





when in substitution for u is made in terms of x, y, z. From the former we have 



a gp np' ^- = g-J- + 

 9 - C p- np' = 



, 

 + 



, 8(5 



? dx 



a? , 



dydz 



and from the latter 



