378 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
and thence writing 
i (® + yi), yi)> for æ > y> z > 
we find the nodes 
0 = f, y = ^i), (x = f, y = -\ i), (x=-l, 2/ = 0); 
the first and second of these are acnodes, or the curve has a pair of imaginary 
acnodes ; the third is a crunode ; and to find the directions at this point, if in the 
equation of the curve we write x — 1 for x, the equation becomes 
yi _ 2y 2 (x — 2) (3x — 20) + (x — 2) (9x — 10) a? = 0 ; 
the lowest terms therefore are 20 (— 4y- + x 2 ) = 0 ; and we have y = + ^ (x + 1) for the 
equation of the tangents at the crunode. 
y = 0 gives a? = l, x=^ and (as a twofold value) x — — 1, which belongs to the 
crunode. 
x — 1 gives y 4 = 0, or the line x = 1 is a tangent of four-pointic intersection. 
x = 0 gives i/ 4 — 34y 2 +1=0, that is y n - = 17 ± 12 V2, or y = ± (3 ± 2 V2). 
75. The curve has a pair of asymptotic parabolas, and taking for the equation of 
one of them 
(y — x V3 + ßf = — % 2 - (a? + a), 
this gives 
y = ¿c V 3 — /3 + V — 4^- (a; + a), 
?/ 2 = 3æ 2 — 2/S a; V3 + /3 2 
and thence 
-¥(« + «) 
which agrees with the value of y 2 in the envelope as to the terms ¿c 2 , and by 
properly determining a, /3, it may be made to agree as to the terms x and x%. 
We have in the parabola 
y 2 = 3x 2 — 2ßx V3 + ß 2 
— ^-X — ^-CL 
and in the curve 
y n - = 3x 2 — 20# +17 
+ V — 32 [2x Vo; — 4 V«}.