242 Mr. Greatheed's new Method of 



where X is a function of x only, Y of y only, and P and Q 

 are functions of both x and y. This may easily be reduced 

 to the form 



dz v dz ^ ^ 



-=- + X -v- = P* -f Q, 

 dx dy 



for 



if v' = / -=^ , Y -yr = -7-7. I shall then consider the 



y J Y cty rfy 



latter equation. 



Let it be put under the form 



g + (x4-p)_ Q) 



and treated as a linear equation between two variables z and x, 



/(xA-p)dx x' ^- -/Pdx 

 The integrating factor ise d y = e d v . s , 



supposing yX d x = X'. The effect of the first part of the 



X'— . 

 factor, namely, e d y is, as has before, been stated, to change 



functions of y into functions of z/+X', and it affects, not only 



d z 



-= — and z 9 but also the other part of the integrating factor, 

 a x 



namely, s ~f Fdx 9 provided P contain y; therefore?/ in P 

 must be changed into y + X 7 before the integration is per- 

 formed, and afterwards, y is to be restored. 

 The equation will then stand : 



= e *» t J Q. 



The first side is the total differential coefficient of e ~~f Pdx z 

 (y + X 7 being substituted for y) with respect to x ; and the 

 second side is the similar differential coefficient of some un- 

 known function of x and y. It would require the solution of 

 the original equation to exhibit this function, but this diffi- 

 culty is avoided by integrating the expression on the second 

 side by the common rules. It remains to integrate both sides 

 with respect to x, add an arbitrary function of y, divide both 



x '~ . -fPdx 



sides by § d y (that is, change y into y —X') and by s , 



and then the value of z is obtained. 



I shall adjoin a few examples, but previously I shall prove 



the following curious theorem, which is of use in several. 



