ME. A. CAYLEY 01^ TSCHIENHAUSEN’S TEANSEOEMATION. 
577 
C=0. And this being so, we may, without loss of generality, write D'=0, E'=0, or put 
(B, C, D, E)=(f2->(0-T)Q(Q+T)-^f2XB', C', 0, 0); 
so that B, C, D, E are linear functions of B', C', such that C=0. And then substituting 
these values for (B, C, D, E), we find a cubic function of B', C'; so that, putting JB=0, 
we have a cubic equation to determine the ratio B' : C. 
The foregoing reasoning presents no real ditficulty, but it is expressed by means of a 
very condensed notation, and it may be proper to illustrate it by the case of the quadric 
function Considering the equations 
y=. 2 ('ki^-\-ii)x'-\r(l — 2 (^ 1 ' — X)z',. 
Z— 2(yK — ^ v^'jz' , 
these equations, if the expressions for w, y, z had been divided by would 
have given 
Hence they actually do give 
or if 
they give 
But if 
then 
so that we have 
C(f-\-y^-\-z^ ■=.(). 
— v) : 2(h>-\-iJij) 
=2(?V|«,+i;) : . 2(p_x) 
=2(vX — : 2(i/jv-[-'k) : 1 — 
w : y : z=l+X^—y;^—v'^ : 2(Xyj-\-v) : 2(vX—iJu), 
which is the same result as would have been found by writing y'=z'=0, and which 
comes to saying that w, y, z are not independent, but are connected by two linear rela- 
tions. 
The equation Det. (n-f-T)=0, written at length, will be 
a , h — r, g+(7, I -\-X 
b j ^ ~S'> 
g — <^5 f + f, C ,11+^ 
1 — X, n—v, p 
= 0 , 
where X, o-, r are the arbitrary constituents of the skew matrix; or developing, this is 
a 
h 
g 
1 
h 
b 
f 
m 
g 
f 
c 
n 
1 
m 
n 
P 
