the Development of a Factorial. 183 



combinations two and two with repetitions of the Greek letters 

 a, ft, and these letters appear in each term in the alphabetical 

 order. Each such combination may therefore be considered as 

 derived from the primitive combination a, a by a change of one 

 or both of the a's into ft ; and if we take instead of the mere 

 combination oc, ex. the complete first term u—au — b, and simul- 

 taneously with the change of the a of either of the factors into 

 ft make a similar change in the Latin letter of the factor, we 

 derive from the first term the other terms of the expression on 

 the right-hand side of the expression. It is proper also to 

 remark, that, paying attention to the Latin letters only, the 

 different terms contain all the combinations two and two without 

 repetitions of the letters a, b, c. The same reasoning will show 

 that 



a?— a x—b x—c x—d=x—u x—ft x-—y x—8 



. V*> A 1, S~| x— a x— ft x—y 

 la, b, c,d\ x 



, \*i ft7 1 X — OL X-ft 



la, b, c, dj 2 



+ M 1 



la, b } c, dj< 



x — oe, 

 is 



+ 



where, for instance, 



[_a,b,c, d] 4 ; 



r«,ft 1 



la, b, c, d]. 



(a — a) (a — 6) (a— c) 



+ («-«)(*-&)(|0-rf) 



+ (*-a)(ft-c)(ft-d) 



+ (ft-b)(ft-c)(ft-d),kc. 



It is of course easy, by the use of subscript letters and signs 

 of summation, to present the preceding theorem under a more 

 condensed form ; thus writing 



La v a 2 . . . a r . . . a r+s ->s+i 



where k s , k s ^ x . . # form a decreasing series (equality of successive 

 terms not excluded) of numbers out of the system r, r — 1 . . . 3, 2, 1 ; 

 the theorem may be written in the form 



p r« « 2 ...a p _ 5P+1 "] 



x—a 1 x — a <2 ...x—a p =:b q \ J x— a Y x— ot 2 . t .x—ot p ^ q ; 



but I think that a more definite idea of the theorem is obtained 



