38 



PROFESSOR A. R, FORSYTH ON THE INTEGRATION OF 



consequent variations of z when x, y, u, vary, we write p and q for dz/dx, dz/dy, 

 respectively; and we adopt the notation of 3. The new expressions for the 

 variations of v, I, in, n, a, b, c, f, g, h, are given in 3 ; those for the variations of 

 a , a l9 2 , 31 A,, ft, &, y , yi , 8 , are given by the equations 



da } 



*g 



r w ~r 



'-dx = >''> + ''^' 



= 7 + 'UP, 



da.?. 

 ~ 



dy 



da z 

 dy 





dy 



dy 



du 



du 



da. 2 

 du 



du 

 dz_ 

 du 



dz 



10 



du 



dy 



_ 

 du 



rf^ 



H 1 



du 



_ rfg 



dz 

 fa 



dz 



V J 



The following relations subsist among the derivatives of , . . . , 8 , free from 

 derivatives of order 4 and from derivatives with regard to u, viz., 



dx 



~dx. 



dx 



dy 



da^ 



~dy 



rlfti J- rJfti 



dy 

 da. 



dy 

 da, 



da } 



dx 

 da; 



dx 



dx 



dx 



dy " dy 



d^ dfiz 



dy * dy 



dy 



dx 



