LAPLACE'S THEOREM. 115 



we deduce y=F(z), 



dz dz ' 



and finally, 



. 



This is called Laplace's Theorem. 



140. Lagrange's Theorem may of course be deduced from 

 Laplace's, by putting F(z) = z. But Laplace's theorem may 

 also be deduced from Lagrange's, thus : 



In the equation y = F{z +ac<j> (y)} .................. (1), 



put z + x$(y}=y', 



then y=F(y'), 



thus tf = z + x<f>{F(y')} ..................... (2), 



and f(y) becomes f{F(y'}}. 



Thus we are required to expand f{F(y'}} in powers of x, 

 by means of equation (2). But this is precisely what La- 

 grange's Theorem effects, the complex functions f{F(y'}} and 



{F(y}} taking the place of the simple functions f(y'} and 



141. It must be remembered, that in quoting Maclaurin's 

 Theorem, which serves as the foundation for those of Lagrange 

 and Laplace, we ought strictly to have used it in the form 

 given in Art. 95, with an expression for the remainder 

 after n + 1 terms. That expression for the remainder however, 

 becomes so complicated in this case, that we have not referred 

 to it. The investigation of Lagrange's and Laplace's Theo- 

 rems must be confessed to be imperfect, since the tests of the 

 convergence of these series, which alone can justify our use of 

 them as arithmetical equivalents for the functions they profess 

 to represent, are of too difficult a character for an elementary 

 / work. The advanced student may consult Moigno's Lemons 



12 



