206 
ME. A. CAYLEY ON THE DOEBLE TANGENTS OE A PLANE CTEYE. 
-.Ve'-dd!-d!d-^V 
j^cid^Vd!-\-dd^d!V^da! 
-a" 
IV=«( We^d'd-\-d!'c 
^^^cd’e-VUl-d’c-d'b 
-f/( a!'a 
-^9 ^ 
which are of the form 
I=« 0 +5(12)+/(13)+^(14), 
II=:«(21)+J 0 4-/(23)+^(24), 
III=«(31)+5(32)+/ 0 +^(34), 
IV=«(41)+5(42)+/(43)+5^ 0, 
where (12)= — (21) &c., and the equation 
I«+IB+III/+IV^=0 
is consequently satisfied. 
23. The expression 
) 
(^1) 
(42) 
(43) 
leads to 
III=- 
a , 
b, 
d 
— 
a , 
c, 
d' 
— 
a, 
d , 
d 
— 
f'- 
a! 
c, 
f 
b. 
d, 
d 
b, 
e , 
d! 
b, 9 , 
b' 
a', 
b\ 
d' 
a!. 
c', 
d’^ 
a!. 
d’, 
d' 
a” 
III = - A^(D^H - 5D^Hi + lOD^H,), 
and consequently the equation of the curve which by its intersections with the tangent 
determines the tangentials of a point of a sextic, is 
D^H - 5D^Hi + 10D^H2= 0. 
24. In the general case of a curve of the order n the matrix is 
( •• CLn-d • 
cIq, di, d,2 . . ; dl, d[ . 
where, in analogy with what precedes, 
=(X, Y, Z)” 
a, =(X, Y, Z)"-'(^, y, z\ 
<-2 ) 
«'n-l 
^n-2 
«„_, = (X,Y, Z) {x,y,zf-\ 
^ 
and similarly for the accented letters, so that 
= 0 is the equation of the curve ; 
=0 is the equation of the fii’st or (^^ — l)thic polar ; 
— 0 is the equation of the last or line-polar, or what is the same thing, since (a:, y, z) 
is a point on the emwe, the tangent at this point ; 
a„ = 0, the condition which expresses that (x, y, z) is a point of the curve ; 
