﻿378 Mr. J. J. Sylvester on two remarkable Resultants arising 

 which, of course, may be indefinitely extended. Thus, ex. gr., 

 when n = 3, the resultant is 

 (^) 3 (a 2 -^)(^-0(6 2 - C 2 )(a 4 + A 4 + c 4 -2a^-2aV-26V) 2 . 



The above investigation leads as a corollary to the following 

 arithmetical theorem. 



Call 1.3. 5... (2x-l) = Q* and 1 = Q . Then 



2x %i +2*(2*-2) %* + 2*(2*-2)(2*-4)%* + ... 



+ M2*-3)---2-£ = (t + | + ^- + 2^K 

 _E#. gr. If a? = 4, 



8^^+8.6.^ + 8.6. 4.^ + 8. 6. 4. 2. ^ 



= 60 + 36 + 32 + 48 = 176. 



So, too, 



3.5.7 + 1.5.7 + 1.3.7 + 1.3.5 = 176. 



The value of u n , i is, of course, II(i — l)Q»_i. 



There is a more elaborate system of 2n equations, the result- 

 ant of which can be made to depend on that of the system of n 

 equations just ascertained. Thus, take 2n = 6, and consider the 

 system 



ax + by +cz +dt +eu +fv; x + y +z +t +u +v; 



ax 5 + by 3 + cz* + dtz + eu*+fvZ', x 3 + y* + z s + t* + u* + v* ; 



aaP+bf+c^ + dP + euZ+fifi; ^+^+^+* 5 +^f^j 



the order of the resultant of this system in the letters a.b.c^ej 

 isobviously 1.3.5(1.3 + 1.5 + 3.5). 



Now pair the six variables in every possible manner ; the num- 

 ber of such pairs is 1 . 3 . 5. 



Let x, y, z, t, u, v be any one such set of pairs. Make 



x+y = 0, z + t=0, u + v=0; 



then the latter set of three functions become zero, and the 

 former three may be made zero with right assignments of x, z, t, 

 provided the resultant of 



{a-b)x +(c-d)z +(e-f)u, 



(a-b)x b + {c-d)z 5 + {e-f)u b 



