42 



Ite 



D 



PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



a _ 7 A A /. JL , ! 



JB 9e 9/ 8w 8w 



+ 



3,3-3 



^7 + " s~ -r / -r 



' 



en 



+ n + A + o - + n + A 



a* 



]) 



n 87 



8 



8. 



Now this is to be satisfied in virtue of the preceding six equations and indepen- 

 dently of the particular values of 8 , . . ., 88 , belonging to any integral of the 

 original equation. Hence the equation must be expressible in the form 



A t {(XX) + ...}+ X, {(XY) + ...}+ X 3 {(XZ) + . . .} 

 + \, {(YY) +...}+ A fi {(YZ) + ...}+ A 6 {(ZZ) +...}, 



where A 1} . . ., A G , are indeterminate multipliers; the conditions sufficient and neces- 

 sary for this are 



0) - - 



- 0, 



G + 



(J G + 



- A( J-G + 





- (iG + 

 + G 



37- + 



o 



- + A 2 (AG 

 + tf) 







- - A^ - 0, 



and 



_8E_ 



8a 



t-') 9 f|-& 

 -ff- + A-g- 



+ ^) ?r ~ i 



C5Cj 



^T^ 



oB 



= 0, 



^_ _ S l-.- 



/3J 5i> W l 



8E 



. A _8E_ 

 ^ 



(YZ) 



8E_1 

 8 3 J 



8E 



3 3 



= 0. 



22. This system must be rendered complete by the addition of the JACOBI-POISSON 



