114 LAPLACE'S THEOREM. 



and thus 



/(y) =/(*) + *$ (*)/' W + 



This result is called Lagrange's Theorem. 



1 39. Suppose y = ^{0 + x$ (y) } ; 

 required the expansion of f{y) in powers of a;. 



Let stand for z-fx<f)(y}; then 



therefore ^ 



ax 



,, r dy dt 



therefore ~ = 



dz , 



Hence fi 



From this, in the same way as in Art. 138, we deduce 

 that 



where u =f(y)- 



If we make x =0 in the equation 



