BiNOMiAi. Theorem. 131 



For, assume that we have tested the above laws in the case of 

 the product of a certain number of factors, suppOvSe «, and have 

 found them to hold true. 



To facilitate the discussion we will represent the n second terms 

 of the binomial factors hy a^, a,,^^ a^, a^, . . . a,,* instead of 

 a, b, r, d, etc., and accordingly the product of the n binomials 

 (x-\-a^)(x-\-a.,J(x-\-a^)(x-\-a^) . . . (x-\-a„_^)(x-^a„) 

 =:r"4-r^iH-«2+^;i+^4 • • -\-(^n}x"~^ 



-\-(a^a^^-\-a^a^-]rCiya^-\r . . -\-a„_^a„)x''~'^ 



-\-(a^a2Ci-f,-\-ci^^a.ya^-\-a^a<2^ar^-\- , . ■\-a„_^^a„_^a„)x"~^ 

 + . . . -\- a ^a ^a ^a ji^ . . a„. 



In order to abreviate this expression it is convenient to let 

 Pi equal the Jirs^ parenthesis, or the sum of all the different 



second terms of the binomial factors. 

 P2 equal the second parenthesis, or the sum of all the different 



products of the second terms of the binomial factors taken two 



at a time. 

 P3 equal the third parenthesis, or the sum of all the different 



products of the second terms of the binomial factors taken three 



at a time ; and so on. 

 P„ equal the n th parenthesis , or the product of all the second 



terms of the binomial factors. • 



With these abbreviations the second member of the above 

 equation reads 



x" + PiJt"-i+P2Jt-"-2 + P3;t-"-=^+ . . . +P„. 



Multiplying this expression, which represents the product of n 

 binomial factors, by a new binomial, .r+a„^ j, we derive the fol- 

 lowing result for the product of ;^-f i binomial factors : 



+ rPa+««tiP2;-^"~'+ . . . +««-iiP..- 



It is seen from this result that the law of exponents still holds. 



For there are n-\-\ binomials and the exponent of x begins, in 



the first term, with n-\-\ and decreases continually by one through 



the remaining terms until the value zero is reached. 



*This notation presents many mechanical advantages. It must not be supposed, 

 however, that there is any relation subsisting between ii-^ and a^ or any other two of the 

 .symbols ; they are as independent as distinct letteis. 



