RESULTANTS. 141 
The three equations of T~‘ and the first one of T, form a 
system of four simultaneous linear equations ; hence 
A 
—-—2x Ai Az As 
p 
A 
—_ - Bi Bs, WB: 
sat Rca rhodes bea (13) 
A 
SS 4 CQ Co Cs 
P 
— Pir © fi nh 
This equation expresses the relation between «, y, z and «,. 
Solving this equation for x, we get 
lz y z O | 
ae Be Co, ar 
PP1 Ae =| 4, BO Als 
A3 Bs Cs i} 
In like manner we get similar results for y, and z,; thus 
ocean 27h enacane (0 ce DE ra oO 
|Ar Bi Ci a Ane Bima Gy is 
P(x A Yo = | Ao Bo Co j32|? PP1 A me = Az Bo C2 j33|° ( 14) 
|As Bs Cs fe As Bs Cs 7s| 
When the determinants are expanded, A divides out of both 
sides of each equation. 
176. Decomposition of the Normal Form into Factors. As 
an illustration of the usefulness of the determinant form of 
the resultant, we shall make use of it to deduce a theorem of 
fundamental importance in the theory of collineations. Equa- 
tions (14) show a striking resemblance to the normal form of 
type I, Art. 180, and, though the A’s, B’s, ete., in the two 
forms have different meanings, they suggest the decomposi- 
tion of the normal form into factors. Since 7, expressed in 
the form of (14) breaks up into factors T and T, expressed by 
(11), so T expressed in the normal form of type I breaks up 
into factors T = UV, where the matrices of U~ and V areas 
follows : 
A A! Al A kA! kA” 
U+4=\\— B Bll andV=|l\B eB KBrll. 
GC! Cr C kc KC! 
