182 Mr. A. Cayley on a Theorem for 



3. Kermesome Type : — Monoclinic. 



Kermesome 2SbS 3 -f SbO 3 . This is not the exact formula of 

 the " Kermes " of chemistry — hence the slight alteration in the 

 name. The true position which the kermesome should occupy 

 in the system has yet to be ascertained. It is placed here quite 

 provisionally. 



XXV. On a Theorem for the Development of a Factorial. 

 By A. Cayley, Esq.* 



THE theorem to which I refer is remarkable for the extreme 

 simplicity of its demonstration. Let it be required to 

 expand the factorial x—ax—bx—c . . . in the form 



x—a. x—fi x—y . . . + Bx—a x—/3 . . + Q>x—a . . -f D . . &c. 



We have first 



#— «=#— a + a— a. 



Multiply the two sides of this by x— b ; but in multiplying by 

 this factor the term x — a, write the factor in the form x—/3+/3—b; 

 and in multiplying the term a— a, write the factor in the form 

 x — a -fa— b; the result is obviously 



x—a x—b-=zx—ct x— /3+(a— « + /3 — b)x— a + a — a a—b. 



Multiply this by x— c, this factor being in multiplying the quan- 

 tity on the right-hand side written successively under the forms 

 x—y + y—c, x—fi + ft—c, x—a + a—c, the result is 



x—a x—b x—c=x — a. x—/3 x—y 



-J- (a— «-f-y3— 6 + 7— c)x— a x—fi 



+ (a— a u—b+u—a @-c + l3—b P—c)(x—a) 

 + (*-a){*-b)(*-c), 

 which may be thus written, 



(x — a) (x — b) (x — c) = (x — u) (x— j3) (x — y) 



■ \*> ft 7~] x—a. X—/3+ [«, 1 x—a + [<* 

 la, b, cj 2 la, b, cj 2 la, b, cj 3 " 



Consider, for instance, 



,/9 "1 =u—ao^-b + *—aP--c+~P—bP--c. 



la, b, cj 



is 



Then, paying attention in the first instance to the Greek letters 

 only, it is clear that the terms on the second side contain the 



* Communicated by the Author. 



