  We consider formal power series $ f(x) = a_1 x + a_2 x^2 + \cdots $ $(a_1
\neq 0)$, with coefficients in a field. We revisit the classical subject of
iteration of formal power series, the n-fold composition $f^{(n)}(x)=f(f(\cdots
(f(x)\cdots))=f^{(n-1)}(f(x))=\sum\limits_{k=1}^{\infty}f_k^{(n)}x^k$ for
$n=2,3,\ldots$, where $f^{(1)}(x)=f(x)$. The study of this was begun, and the
coefficients $f_k^{(n)}$ where calculated assuming $a_1=1$, by Schr\"oder in
1871. The major result of this paper, Theorem 7.1.1, generalizes Schr\"oder
[15]. It gives explicit formulas for the coefficients $f_k^{(n)}$ when $a_1
\neq 0$ and it is viewed as an analog to the $n^{th}$ Multinomial Theorem Under
Multiplication. We prove Schr\"oder's Theorem using a new and shorter
approach.The Recursion Lemma, Lemma 3.1.1,which sharpens Cohen's lemma (Lemma
2.4.2) is our key tool in the proof of Theorem 7.1.1 and it is one of the most
useful recurrence relations for the composition of formal power series. Along
the way we develop numerical formulas for $f_k^{(n)} {\rm where} 1\leq k\leq
5$.
