•200 
ME. A. CAYLEY ON THE DOUBLE TANOENTS OE A PLANE CUETE. 
111 = a{—h''d—d’d-{-Vd’^c'd-\-(l!ll ) ( 31 ) 
+5( a!'d-{-b"c~\-d'b--ad’~b'd—db' — d'a') (32) 
+(?( 0 ) ■ 
-^e{--a!’a-\-o!a! ) ■> (^“^^ 
IV= a{ b''c-\-d'b—Vd—dV ) (-^1) 
j^l[-.c^<c-b"b-d'a-\-a!d-\-Vb’-^dci ) ( 12 ) 
+(?( d'a—dd ) 
+4 0 )’ ■ 
whicli are of the form . % jn€t\ \ a^\ 
II=a(21)+J 0 +d(23)+e(24:), 
III=«(31)H-J(32)+d! 0 +e(34), 
IV=441)+J(42)+(^(43)+e 0 , 
where (12)= — (21) &c., and which therefore satisfy the equation 
Ia+IK+IIW+IVe=0. 
12. The equation of the curve which by its intersection with the tangent gi^es the 
tangentials, is 
III=- 
a, 
b, 
d 
— 
a, 
c. 
b' 
— 
a. 
d, 
d 
b. 
€, 
d’ 
b. 
d, 
d 
b. 
e, 
V 
d, 
b', 
d' 
d, 
d. 
b" 
d, 
d', 
d' 
= 0 , 
the degrees of which are ^ r tt q 
in the coefficients of U , o, 
in (^ , ^ , z ) • • • 6, 
in(X,Y, Z) . . . 4, 
in (X', Y', Z') ... 2; 
and it only remains to divest this equation of a factor which it contains, 
X , y , z 
X, Y, Z 
, X', Y', Z' 
which being thrown out, the equation will be independent of (X', Y', Z') and of the 
degrees 
in the coefficients of U, o, 
in (x ,y , z) . . . • 4 , 
in (X, Y, Z) . . . . 2, 
and will in fact be the before-mentioned equation Q=D®H— 3D-H, = 0. 
