ME. A. CATLET ON THE DOUBLE TANGENTS OF A PLANE CUEVE. 
203 
18. Hence writing %=4, we have 
2 
a , 
c' 
+ 8 
a , 
d, 
a! 
+2 
d, 
d 
b. 
c. 
d! 
e , 
V 
e. 
V 
a', 
V. 
c" 
a!, 
d!. 
d' 
d. 
d', 
d' 
2 
a, 
d , 
a! 
h, 
e , 
V 
n. 
d\ 
d' 
and hence 
2 ; 
a, 
b, c' 
+ 
a , 
c , 
V 
+ 
«, 
d , 
d ] 
b, 
c, d' 
d^ 
d 
d J 
d. 
b\ c" 
n'. 
c\ 
W 
n'. 
d\ 
d' 
6 5 5 c 5 q! 
c, d, V 
c\ a!' 
— |— 2 J ) C f CL 
c, V 
h\ c', a" 
>=A^(D^H-3D'H,), 
which is the required equation, 
III=-iA^(D^H-3D^H0. 
19. It is to be added, that the equation for A^DH gives IV=^A^DH; the values of 
II and I are at once obtained from those of III and IV by interchanging {x, y, z) and 
(X, Y, Z). Hence if we represent by |^, 2B, &c. the values which H, D, &c. assume by 
this interchange, we may write 
II=+iA^(M-3B^li), 
III=-iA^(H^H-3D^H,), 
IV = +iA^HH; 
and the identical equation, 
ia+m+iii(^-l-iVe=o, 
gives therefore 
-iai.U+i(M-3M,)HU-i(D^H-3D^H,)i3T+DH.T=0, 
which is of itself sufficient to put in evidence the property that the curve D^H — 3D^H, = 0 
gives by its intersections with the tangent DU=0, the tangentials of the point (x, y, z). 
The last-mentioned equation is the equation for a quartic corresponding to Mr. Salmon s 
equation 
-l.U-l-i».HU-iDH.BT+H.T=0 
for the cubic U=0. 
20. It is worth while to give the investigation of the equation for the cubic; the 
matrix is 
( a, 
h. 
c ; 
d , 
y ) 
c. 
d-, 
y . 
d 
b\ 
d’, 
2 E 2 
d\ 
y 
