d , Y, X being all of them expressed as functions of x, y, a, the expression 
on the right-hand is a complete differential, and we have 
b= [^(Ydx-Xdy); 
dz 
that is, the integral b is determined by a quadrature. 
27. Thus, in the example No. 22, 
dX dY dZ_ 
dx dy "I" dz ’ 
and a value of the multiplier is =1. Supposing that the given integral is a = x + y + z, 
then ^=1, and we have accordingly 1 as the multiplier of the equation 
y (a— 2x — y)dx + x (a — x — 2y) dy = 0, 
viz. this equation is integrable per se. Supposing the given integral to be b = xyz, 
then ^=xy, viz. we have — as the multiplier of the equation 
dz xy 
Cx ~ Xy ) dx + {y~ Xy ) dy = °’ 
and we thus in each case obtain the other integral as before. 
28. The foregoing result may be presented in a more symmetrical form by taking 
in place of x, y any two variables u = u (x, y, z), v = v (x, y, z). 
Supposing the integral a known as before, the system then is