432 Mr. J. J. Sylvester on Multiplications, ^-c. 



then it is easily seen that 



{"e, + fle, + • • • + %)" = {% + %,+ •■■+ %„_ , + «„)" 

 2(-«e,+«0^,+ ... +«0j'=S(fl'^^ + «^^+ ... +«^^^_^-«J" 



+ 2(-«^. + «^,+ ...+«^^_, + «„)\^ 



&c. = &c. 



(-«9,-«02 • • • -«9„_,-«0„)"=(-«'^,-«0o • • • -«0„_, -«„)". 

 Hence it is apparent that when a^^= 0, the right-hand side of 

 the so-called principal equation spontaneously vanishes ; it will 

 therefore always contain a„ as a factor, and by parity of reason- 

 ing it will contain every one of the quantities o,, a^, . . . a^ 

 as a factor^ and will consequently be equal to the product 

 flTp ffg, . . . ttn multiplied by a numerical factor, which by making 

 «„ «2J • • • ^?» ^^ch. equal to unity, is readily seen to be 

 2" X (1 . 2 . 3 . . . «) 



(2" being the sum of the numbers of terms in the (n + 1) groups) ; 

 or if we please so to say, to 3 . 4 . 6 . . . (2/i). Q. E. D. 



Conclmion. 



If n is odd and be called 2m -f 1, we have 



4.6.8... (2n)«i .a^...a„ 



= (fl0, + fle„+ ... +«J»-S(-«0, + fle^-f ... +«0j" 



+ X{-ay-ae^ + ag^+ - • • +a0„)" + &c. 



and if n be even and be called 2m, we have luat 



4.6.8... (2n)K.fl2...«J 

 = K+«e2+ ••• +«0„)"~S(-a9j + «0,+ ... +«e„)" 



+ i(-r2:(-%-ff0,. ••-«e„. + ^'0,„+, + '^e,„ .,, + ••• +«0j"; 



where, it should be observed, that the last term is made up of in- 

 teger parts, notwithstanding the presence of the factor i, which 

 factor may be construed as only ser\ing to denote that, of any 

 pair of complementai-y linear functions of those which enter into 



