Method of Integration. 85 



arrive at an equation of the first order ; but in equation (53), 

 if for 2"+\ we write ;/^2"+\ where ni denotes an odd number, 

 there the method may be used to reduce an equation of the 

 order inT^ to one of the order ;//. 



(Additional Note received January 2^th, iSqj.) 



In the former portion of the paper I suggested, for 

 reducing and integrating certain partial differential equa- 

 tions, a method which did not require a knowledge of the 

 general law of integration of equations of that order. In 

 this additional note I wish to point out that if in line 12 of 

 the previous paper we substitute subtraction for addition we 

 shall still obtain integrable forms. As a particular example 

 consider (i) and subtract from it (2) then we shall obtain 

 the equation 



dh _ dh _ dH _ dh 

 dx^ dxdy dif dydx 



but this may be written in the form 



d rdz _ dz\ ^ _ iJ^/dz _ dz\ 

 ix\dx dy) dy\dx dyf 



dx 



of which the integral obtained in the ordinary way is 

 dz dz , , 



By means of (8) and the present equation, equation (9) 

 may be obtained in the usual manner. 



Next consider the equation of the 4th order (13); if from 

 this we subtract the auxiliary equation (14), we obtain a 

 result which may be written in the form, 



dx^^'df'^ ^^^^ 



wherein 



_ dH _ dh_ _ 

 d:)^ dy^ 



