282 J^ew Algebraic Series, 



From the preceding it follows, ttiat 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 (function of 

 a) be put for the first series, or 



fa=]+aY+a{a-{-k)-f^+ he. 

 ihenfb=l + bY+b{b+b)^-{- he, 

 and also/(a+6) = l+(o+6)Y+(a+&)(a+6+A:)p2+&c. 



Our theorem then will be of the form 

 fa.fb=f{ai-b). (I.) 



From this notation it follows evidently thatyb=l. For 

 making b=o, a=\,,fo=fl=\. 



If in equation (I) b be changed into &+c, it will become 

 fa.f{b-\-c)=f(a4-b-{-c) ; but from the same equation /(6-f- 

 c)—fb .fc, whence by substitution we have/a .fb .fc=f{a 

 4-6-f-c) : Again changing c into c+d, we have, in the same 

 m&nner,fa . fb . fc. fd=f{a-{-b-\-c-\-d) 



Whence in general 

 fa . fb . fc . fd+ &c. =f{a + b+c+d-h . . . &c. 



That is, the product of any number of series, of the form 

 of the series fa ; differs from each other only, in having ft 



successively changed into 6. c, d or to a series into 



which a.b c, d . . . he enters in the same manner that a 

 enters into the first series, b into the second, &c. 



If in this last equation, the quantities a,b,c,d. . ... &c. 

 be supposed equal, and their number equal to m, then 

 (faY==fma. (H.) 



That is, any entire and positive power m, of the series 

 Ja, is a series into which ma enters, in the same manner as 

 a in the first. 



From equation (I) we have fb . fc=f(b-{-c), assuming 

 6-|-c=a, then c=a — 6, and by substitution y6X/(a — 6) =/«, 



fa 

 whence -^v =/(a — 6) . (III.) 



