84 



PROFESSOR A. B. FOKSYTH ON THE INTEGRATION OF 



a = 



7 1 S $1 



7 i (\ O 



g - iSjSjW 



All these forms correspond to the respective expressions obtained in 47 for the 

 potential-equation. 



52. Account must now be taken of derivatives with regard to u, and the equations 

 to be satisfied are 



1 di< 1 M 1 dm 1 dn dz 



n dti (j dn f du c du du 



Taking the first, we have 

 so that the equation is to be 

 Now, denoting 



= 1 + xp' + yq' = A, 



/ 



P 



f (.1 



~r 



dy 



and so for 8/, S 3 ' viz., S/ is the same as Sj except that p' and 5' replace p and g, and 

 so for S' we have 



^"S 2 } 



~ 

 du 



and therefore 



28* ^ 



Now 

 so that 



SA = pp' -f qq = 0, 



8 (AS 3 w,') = 

 Hence the equation in w will be satisfied if 



28 



the condition that we may have 



dv 

 du 



dz_ 



du ' 



