72 Mv. Jarrett on Algebraic Notation. 



S„, S'\ a„ 62„ _ 1 = ii . S„, a„„ i+b^. S,.. a,„^,,. + b,. S,„ a,„ ^ „, 3 + &c. 



QC CO CO !^ 



and S„S";,a i.,, = 62 • S,„a„„, + h^.S,„ + ,,o + Oc • S„,a„^2,3 + &c. 



X 00 



23. We frequently need a symbol to denote the sum of all 

 the combinations that can be formed of n quantities taken m at 

 a time. Let then the letter C be taken as an abbreviation of the 

 word combiuatiotis, and C " '(a,) may be used to denote tlie snm 

 of every possible combination of « quantities, of which the j* is 

 a„ and of which we are to take m at a time. 



24. Theorem. If b is independent of r, then shall 



Cr" {a..b) = b"'.Cr"{a,). 



For C"''" (a^b) denotes the sum of a series^ each term of which is 

 a product of m quantities, and into each of which quantities b enters as 

 a multipher; and C,."''"(a) denotes the sum of a series, each term of 

 which is a product of the same m quantities deprived of their multiplier 6, 

 and hence, by Art. 6 and 13, the truth of the proposition is obvious. 



25. Theorem. ^^r^ = Q'--". (-) 



P- (a.) 



For, the numerator of the left side of this equation consists of every 

 possible combination of n quantities taken m at a time ; and hence that 

 side of the equation consists of a series of fractions, in which the numerator 

 of each is unity, and in which the denominators are formed by taking away, 

 in every possible hianner, m of n given quantities, and will therefore con- 

 sist of every combination of those n quantities, taken m -n at a time. 



