Method of Integration. 87 



the letter c denoting some quantity as yet undetermined ; 

 then by addition we obtain an equation which may be 

 presented in the form, 



d /dz dz\ d/a\Jz dz^ 

 dx\dx dy) ~ f^\ dy dxj' 



or as it may be written 



dx\dx dy) dy\c dy dxj' 

 now let c be so determined as to make the coefficients of 

 ^ on each side equal, then the equation may be written in 

 the form 



dv dv 



dx-%f (^9) 



wherein 



_ dz dz 

 dx dy^ 

 and also 



d' 



the last equation gives for c, the values -\-a and -a. Inte- 

 grating (59) and giving for v its value we have 



dz dz I . 



-dx^%='^^y^''y^ 



and substituting for c its two values we have 

 dz dz , 



dz dz , , 



dx dy ^^^ ' 



and the rest of the operation may be continued as usual. 



The method of integration suggested in this paper admits 

 of more general application than has previously been con- 

 sidered, and may be applied to some equations which are 

 not of the general type already considered. For instance 

 take the equation 



f^_ ^ (Pz 



dx^ dy"' dydx 



