FOBM OF TANGENTIAL EQUATION. 375 
In the usual notation this is 
4afc 3 +27/^=0 (26) 
The equation (24) shows that the sum of the xs of the points where any line cuts 
ay 2 =x 3 is proportional to the square of the tangent of the angle which the line makes 
with the axis of x, and the sum of their reciprocals is proportional to the reciprocal of 
the intercept which the same line makes on the same axis. 
Cor. It is evident that similar theorems hold for the curve ay n ~ x =x n . 
(2) Let the curve be x 3 +y 3 — 3axy=0. 
The tangential equations are 
v 4 — (6a 2 cot <p)v 2 — 4« 3 (l-{-cot 3 <p)v+3a 4 cot 2 <p=0, (27) 
v 4 — 6 1 fXyjv 3 + 4 a 3 (A 3 + j^ 3 ) ^ + 3 a 4 X 2 gj 2 =0 (28) 
(3) Find the tangential equations of the cissoid. 
They are 
(2a— v) 3 =27a 2 v cot 2 <p, . (29) 
(2a\+v) 3 + 27 ay v=0 (30) 
(4) Find the tangential equation A x for a cubic in its canonical form — that is, referred 
to its three chords of inflection as axes. This question is solved by supposing the coeffi- 
cients in the equation (12) to vanish, except a w a 3 , b . 2 , d 3 ; then equation (14) becomes 
the conic 
a 3 i>X+2fry=:0, (31) 
and the curve A, for the Hessian of the cubic is 
3(a 0 b 3 d 3 f%vx=(a 0 b 3 d 3 +2b 3 2 y, (32) 
a curve which has double contact with the former. 
(5) Find the equations of the curves A 1? A 2 , A 3 for the trinodal quartic 
(a, b, c,f, g , li$jc~ x , y~ l , z~ x f. 
A,=^ 2 %X-/^+cv)=0, (33) 
A 3 = y?v(a\— hgj+gv) — Q, (34) 
A , = («, b , 6, -/, 2 g, -70£X, p, >T=0 (35) 
(6) The points where the curve A m intersects the line CN may be found as follows : — 
If a variable point moves along the line CL the envelope of its polar curve of the mt h 
degree with respect to U will be a curve of the degree 2m(n— m— 1) which will cut CN 
in the required points. Similarly the points where it cuts CL may be found. 
3 H 
MDCCCLXXVII. 
