350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
377 
We may fix the absolute magnitudes of the coordinates by writing x + z = 1; and the 
equation then becomes 
(a? 2 + y 2 ) (1 — x) + 4& {(\ — v) x + fxy + v} = 0. 
The origin is at the asymptote-point, or intersection of the imaginary asymptotes ; the 
equation of the real asymptote is x = 1 ; that of the imaginary asymptotes is x 2 + y 2 =o. 
If x and y are ordinary rectangular coordinates then the pair of lines represented 
by this equation will be an indefinitely small circle, and conversely, if the Asymptote- 
Point be an indefinitely small circle, then x and y will be rectangular coordinates ; 
and we may without loss of generality assume that this is so. 
73. The equation of the envelope is 
v / i(x + yi) + \/\(x - yi) + \/z = 0 ; 
this gives successively 
(x + yi) + (x — yi) — \/z = — V2 \/ X 2 + y 2 , 
x + z- \/a? + y 2 = 2\/z{s/%(x-\-iy) + (a? - iy)\, 
(x + z) 2 + a? + y 2 -2(x + z) Va; 2 + y 2 = 4z(x + Vx 2 + y 2 ), 
(x-z) 2 + x 2 + y 2 = 2(x+3z) Vx 2 + y 2 , 
{(x — z) 2 + x 2 + y 2 } 2 = 4 (x + 3z) 2 (x 2 + y 2 ) ; 
that is 
(x -z) 4 + 2 (x 2 + y 2 ) {(x - z) 2 - 2 (x + 3z) 2 } + (x 2 + y 2 ) 2 = 0. 
Putting z =1 — x, and therefore x — z = 2x — 1, and x + 3z = 3 — 2x, this becomes 
(2x -iy + 2 (x 2 + y 2 ) {(2x - l) 2 - 2 (2x - 3) 2 } + (x 2 + y 2 ) 2 = 0, 
that is 
(x 2 + y 2 ) 2 — 2 (x 2 + y 2 ) (4r 2 — 2Qx +17) + (2x — l) 4 = 0, 
which may also be written 
yi _ 2y 2 (3x 2 — 20a? + 17) + 9a? 4 + 8a? 3 — 10a? 2 — 8a? + 1 = 0, 
or, what is the same thing, 
y 4 — 2y 2 (a? — 1) (3a?— 17) + (a? — 1) (9a? — 1) (a? + 1) 2 = 0, 
for the equation of the envelope. The solution of the equation in y gives 
y 2 = (a? — 1) (3a; - 17) ± (2a? - 3) V-32(a?-l). 
74. The original curve v / a?+\Xy-(-v / 0 = 0 had the three nodes 
(2, — — \), (— 2, — |), (— 2) ; 
C. V. 
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