ME. A. CAYLEY ON THE DOUBLE TANGENTS OE A PLANE CUEVE. 
201 
13. Write for shortness, 
it is to be shown that 
X , y , z 
X, Y, Z 
X', Y', Z' 
W, 
III= 
14. To effect this I remark that we have identically 
a, h, d = A^H ; 
5 , c , V 
a', h\ d' 
and I proceed to operate upon this equation with D=XB^+YB^+ZB^. 
I notice that 
a, i, c, d, e; a', h\ d, d ' ; d\ h", c" 
are in regard to {x, y, z) of the degrees 
4, 3, 2, 1,0; 3, 2,1,0; 2, 1,0; 
or what is the same thing, since for the case in hand %=4, of the degrees 
; w— 1, w— 2, ..; w— 2,^—3, . 
and we have 
J)a=znh, m-{n-l)c, . . T>d={n-l)h’, By={7i-2)c\ . . J}d'={n-2)b\ 
15. In the determinant 
a, h, d 
5 , c , y 
y, d' 
d, 
, =A^H, 
the degrees of the terms (other than each top term, the degree of which is higher by 
unity) in the several columns are n — 1, n — 2, n — 2; if then we operate on the deter- 
minant with D, and as regards the top terms we write 
D« =-h -\-{n—l)b , 
jyh =c +(w— 2)(?, 
J)d=y+(n-2}y, 
we have in the first place a term 
which vanishes, and next the terms 
(n-l) 
b, 
b, 
d. 
c, 
c, 
b', 
y 
y 
b, b^ d \ -\-(^n—2'j 
a, 
c, 
d 
+ {n~2) 
a, 
b. 
y 
c, c, y \ 
b, 
d. 
y 
b. 
c, 
d 
y, y, d'\ 
d, 
c', 
d' 
d, 
y. 
¥ 
2 E 
MDCCCLIX. 
