214 Mr. J. J. Sylvester on a ?iew Class of Theorems 



shall coexist; for then two equations between x, y, z 9 of which 

 lx + ?ny + nz = will be one, will suffice to make U and V each 

 identically zero. Hence we have the following theorem : 



I {\XJ-hpY+(lx + my + 7iz)t 



X/j, xyzt 



is a factor of the resultant of 



P = Q = lx + my + nz = Q. 



A comparison of the orders of the resultant and the deter- 

 minant shows that they must be identical, d-ci-pres, of a nu- 

 merical factor, which, if the resultant be taken in its general 

 lowest terms, may no doubt be easily shown to be unity. 



As an illustration of our theorem, let 



P = xy -f yz-\-zx 

 Q = cxy 4- ayz + bzx. 



Then 



{*.F+pQ+(lx+-my+nz)t} 



xyzt 

 f A-f tyo \-\-bp I 



I A + qu, A + ap, m 

 | \-\-by, X + ap n 



LI m n J 



= rc 2 (A + C[xf + m\\ + V) 2 + P(X + a/*) 2 

 — 2/w( A + fy/,) (A + a\L) ~ 2»z w (A + qx) (A + ^ ) 

 — 2?zZ(A4- ap)(\ + cp) 

 = A 2 {rc 2 + m 2 4 P-2lm — 2?nn — 2nl} 

 + 2\/ji{c?i 2 + bm' 2 + al 2 —lm(a + b)—mn(b + c) — nl(c + a)} 

 4- p, 2 { A 2 + Z> 2 ?w 2 -f a 2 / 2 — 2a£/w — 2bcmn — 2canl } . 

 And we thus obtain, finally, 



~| {AP + i aQ4- (lx + my + nz)t} 



=.{n 2 + m 2 -\-l 2 ~-2lm-2mn — 2nl) 

 x [fin 9 - 4- & 2 m 2 4- a*P—2ablm — 2bcmn — 2canl) 

 — {(c?i 2 -t-b?n <2 + al 2 —lm(a + b)—mn(b + c) — nl(c + a)y 

 = —Mmn{(a — b)(a — c)l+{b — a){b—c)m + (c—a)(c—b)} m 

 Now to obtain the resultant of 



xy+yz + sx—O 

 cxy 4- azx 4- fo^ = 

 Ix 4- wz/ 4- nz = 0, 



