AND ON DEFINITE INTEGRATION. 469 
left-hand side of the equation (3), and Laplace’s 
theorem to the right; then differentiate succes- 
ee 
methods of Maclaurin and Taylor, are evidently in them- 
selves really identities. 
Given z=y+2f(z) to find the value of (not only z) but 
F(z), in terms of y, and ascending powers of «. 
Given 2=F, Sy-te fe) to find the value of (not only 2) 
but F(z), in terms of y, and ascending powers of wx. 
The complete solution of these functional equations, con- 
ducts us to what are called Lagrange’s and Laplace’s theorems, 
which are always obtained by the assistance of the theorems 
of Maclaurin and Taylor. The solution of the second equation, 
as given by Lacroix in his large work on the Differential and 
Integral Calculus, vol. 1, page 279, is remarkable, in conse- 
quence of its containing the great principle of the convertibility 
of independent differentiations. 
Now, it is stated, by most of the modern writers on the 
Differential and Integral Calculus, that the theorems of 
Maclaurin and Taylor are only particular cases of those of 
Lagrange and Laplace. 
With great respect, however, for the abilities, and opinions 
of these writers, I must say, that, there does not appear to 
me, to be any just grounds for making such an inference. 
I grant, that if in the first equation we make f{(z)=1, its 
solution will contain the developement of F(y+.). And if 
in the second equation we make f(z)=1, and F(z)=z, its 
solution will contain the developement of F, (y+2). But 
what does this lead to, it only shows to any person who will 
take the trouble to examine the solutions, and then to retrace 
3Q 
