IVIALLET. 341 



involves some algebraical work which we will give, since as we 

 have intimated Mallet himself omits it. 



Let there be r quantities a,h,c, ... h. Suppose x^ to be di- 

 vided by (x — a) (x — b) (x — c) ... {x — h). The quotient will be 



x'-" + H^ x^-' + H^ x^-'^ + . . . in infinitum, 



where E^ denotes the sum of all the homogeneous products of the 

 degree r which can be formed from the quantities a,h,c, ... Ic. This 

 can be easily shewn by first dividing a;^ by x — a] then dividing 



the result by x — h, that is multiplying it by a?~M 1 J , and 



so on. 



Again, if ^ be not less than r the expression 



x^ 



(x — a) (x—b) ... (x — k) 



will consist of an integral part and a fractional part ; if ^ be less 

 than r there will be no integral part. In both cases the fractional 

 part will be 



ABC K 



X — a x — b x — c x — k' 



where A = 



a" 



(a — b)(a — c),.. (a — k)^ 



and similar expressions hold for B, C, ... K. Now expand each of 



A B 



the fractions , 7 , . . . according to negative powers of x ; 



X "~" a X ~~ 



and equate the coefficient of £c~*~^ to the coefficient in the first 



form which we gave for x^ -^[(x — a) (x — b) ... (x — k)]. Thus 



Aa'+BU+ Cc' + ... + Kk' = E, 



?-*•+«+!• 



Put m for ^ — r + ^ + 1 ; then p + ^ = m + r — 1; thus we may 

 express our result in the following words : the sum of the homoge- 

 neous products of the degree m, which can be formed of the r quan- 

 tities a, b, c, ... k, is equal to 



m+r-l 7,m+r- 1 



+ 77 TTI ^ 77 fT+ ... 



(a — b) (a — c) ... (a — k) (b —a) (b — c) ... (b — k) 



