112 On some Equations in Partial Differentials. 

 that is x'-f=f{y-x). 



and 



dx 

 'fix 



therefore z=J*'^f{y-x) + ^{y) 



Sometimes we may succeed by partially adopting these 

 processes. Let 



d^z , ^. dH d^z , dz 



or xWz + [xy + 1 )DD'^ +yV)f^z + D'^= 0. 



Now j/D'~=D'j/r-r, 



thei'efore 



xy DD'r = ^DD'j/ z - xViz, yD'^z = D' V- 2D'^. 

 Therefore by substitution 



a:D2^ + xJ)I>'yz + DD'z + D'^yz - xBz -D'z=0. 

 Or (a:D + D')(Dz + D'j/^-s) = 0, 



which may be put under the forms 



(xD + D')(D+3/D')^ = 0; 

 or {D + x-''D'){T>+yB')z = 0. 



Make {T>+y'D')z=u; 



then (D + A-'D')z^ = 0, and ti = x-°'f{y). 



Therefore 



{T>+yD')z = x-^'f{y) = ,-'^^'f{y)=f{y-lx); 



which gives z = b-''!''°' f dx e^'J^' f {y-lx). 



It must be remembered that 



£±.r^D' _ 1 -r_:r( jyD') + 1- {yT>f± &c. 



= 1 ± A-^ D' + 1- ?/D'j/D' + &c., 



the powers being converted into successive operations. 



Sometimes an equation may be reduced an order lower, or 

 to a more simple form of the same order, by these transfor- 

 mations, as in the following example. 



_ d^z , o ^^~ ^ ^ dz „ 



Let j~^ + hx^-j-^ — 2A-p = 0. 



dx^ dx^dy dy 



Make ;? = (D + Aa'2D')m, and the above will reduce to 



A,l3 /i,<3 /,j3 / ],t^\ 



