4.99] 
ON THE THEORY OF THE CURVE AND TORSE. 
91 
thirdly, the points H each counting as 4 cusps. For first consider a generating line 
meeting the curve x in B and the curve x in A ; if we imagine on the curve x a 
point Q which approaches and ultimately coincides with B, the generating line through 
Q meets the torse in the neighbourhood of its cuspidal edge in two points which 
come ultimately to coincide with the point A, and we thus see that A is a stationary 
point on the x (r — 2) curve. 
54. Secondly, observing that the line v is a cuspidal line on the torse, and con 
sidering in like manner a generating line of the x cone, which approaches and comes 
ultimately to coincide with one of the x— 2r + 9 points, we see that this is a 
stationary point on the x (r — 2) curve. And thirdly, any line through a point H meets 
the torse in this point counting 4 times, and in r — 4 other points. Hence considering 
the generating line of the x cone, which travelling along any one of the four partial 
branches of the x curve comes ultimately to coincide with H, 2 of the r — 2 points 
on such generating line come to coincide at the point H; and we have thus the 
point H as a singular point on the x (r — 2) curve; viz. it reckons as a stationary 
point once in respect of each of the four partial branches of the curve x (it must 
be assumed that this is so, but a further proof is required), that is as 4 cusps on 
the x {r — 2) curve. 
55. By what precedes we have 
x (r — 2) (r - 3) = n(x— 2r -1- 8) 
+ 2(2k+ o)(x- 2r + 10)} 
-F 3 {{mx — a— '3ft — 2y — Sv — 4« — 8H) + v(x — 2r + 9) + 4/7}, 
which is the true theoretical form in which the equation tor x (r — 2) (r — 3) was 
obtained by Cremona.