30 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



independent solutions. These can be obtained, by any of the regular methods, in 



the form 



2a 3h + b + f y 21 in n 







But the last is zero, owing to the original differential equation ; and by using this 

 imposed restriction, the second becomes 



a - ->h + I 



.'/ + * 



Consequently the most general solution of the system is 



a -2 



* 



where <1> is arbitrary ; an equivalent of this is 



a - -2h + I a , 



,, + , = e (- + *'> y)> 



where d is arbitrary. 



Similarly a new relation between derivatives of the second order can be deduced 

 by taking the alternative solution p q = of the characteristic equation. 



The rest of the solution proceeds as before when once the system of partial 

 differential equations satisfied by 



$ = 9- (a, b, e,f, <j, /,, I, m, n, v, x, y, z) = 

 is obtained. Now, when p = <j, the relations of identity are 



ill^ da Idij dij 



d,c d;/ ~~ * \di/ ili 



(tl^ dh^ _ Idf df 



iLv " dy ~ ^ (dy " da ' 



df djf _ fdc, (k\ 

 d.v ~ dy ~~~ P \dy " dx)' 



These can be used to eliminate p from the equations particular to the present case, 

 and the latter then become 



da_ da /djL <lg\ -i 



dx dy \dx~-~dy) + ^ 



ML M ( d f_ df\ 



dx dy "\dx " dy)^ ' U 



djl dy_ /dc_ 

 dx " dy "\dx " 



