326 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
2k 
with the like values for g and h; or omitting the common factor - n , we have 
u 
(a, b, c, f g, h) = (e + 0 + 4* 6 + 4v, -0-^, -0-^, -0-^£ 
and thence, taking now x, y, z as current coordinates, the equation of the tangents at 
the node is 
(a, b, c, f, g, li#x, y, zf = 0. 
34. Substituting for A, /x, v the values a -3 , /3 -3 , y~ 3 , and for 6 the twofold value 
— the equation of the tangents at the twofold centre becomes 
J3~T (4/3y - a 2 ), . , . , a 2 /3y (J3y + 2a 2 ), . , .) (x, y, zf = 0, 
which is at once reduced to 
(/3y (/3 - y) 2 , . , . , a 2 /3y (y - a) (a - /3), . , )j {x, y, zf = 0, 
or, what is the same thing, 
{/3y (/3 - y) x + ya (y - a) y + a/3 (a - /3) zf = 0, 
which shows that the twofold centre is a cusp, and that the tangent is 
/3y (/3 - y) sc + ya (y - a) y + a/3 (a - /3) z, 
or, what is the same thing, 
0S-7)f+ (7-«)|+ («-0)i = O. 
35. Writing in like manner A, /x, v = or 3 , /3 -3 , y~ 3 , and 6 for the one-with-twofold 
2 
value = , we find for the equation of the tangents at the one-with-twofold centre 
^2/3 2 y 2 (2/3y + a 2 ), . , . , - a 2 /3y (2/3y + a 2 ), . , (,x, y, ^) 2 = 0, 
which may be reduced to 
(2^ (2/3y + a 2 ), . , . , a 2 /3y (2ya + /3 2 + 2a/3 + y 2 ), . , .j (x, y, zf = 0, 
or, what is the same thing, 
(/3y# + yay + <*Pz) {(2/3y + a 2 ) /3yx + (2ya + /3 2 ) yay + (2a/3 + y 2 ) a/3z] = 0. 
Hence at the one-with-twofold centre the equation of one of the tangents is 
(2(3y + a 2 ) f3yx + (2ya + /3 2 ) yay + (2 a/3 + y 2 ) a/3z = 0, 
or, as this may otherwise be written, 
(2/3y + a 2 ) - + (2ya 4- (3 2 ) ^ -I- (2a/3 + y 2 ) - = 0. 
a p y