282 . New Algebraic Series. 
From the preceding it follows, that if this law holds for 
any term, it will hold for the following term, and as it has 
been proved that the law holds for the four first terms, it 
must therefore hold for every term. 
In order to abridge the above formula, let fa (fanction of 
a) be put for the first series, or 
22 
_fa=!1 $a>+a(a+h)>-+ &e. 
then fo=1-+45-—-+0(b-+b)— 5+ be. 
and also f(a+b)=1 +(a+b) +(a+b)(a+b-+h)=s +b. 
Our theorem then will be of the form 
Sa fo=f(a+b). (1) . 
making b=0, a=1, fo=f1=1 
. 
_ (fay"=fma. (IL) 
That is, any entire and positive power m, of the series 
Ja, 1s a series into which ma enters, in the same manner aS 
a = the first. mea 
rom equation (I) we have fb. fe=f(b+c), assuming 
6-+-e=a, then c=a—b, and by substitution fb X f(a — b)=fa; 
heme F =fa-) » (ILL) 
