90 
ON THE THEORY OF THE CURVE AND TORSE. 
[499 
two coincident sheets; for a single sheet the number of intersections would be 6, but 
for the two coincident sheets it is twice this, or =12. Finally, a point t is an 
ordinary point on the second polar, each of the three branches of x simply cuts the 
surface, or the number of intersections is = 3. 
49. The last table gives at once 
m(r — 2) = 2?i + 4/3 + 7 + 4v + 2.2<w + 4H, 
x (i— 2) = (r — 4) + 2/3 + 37 + 4v + 3.2tu + Sv (r — 6) + 2« (7— 8) +12H + 31, 
which are the true theoretical forms of the equations for m{r — 2) and x{r — 2), in 
which these were obtained by Cremona. 
50. The x (r — 2) (r — 3) points are those points in which the Cremona x (r — 2) 
curve is met 2-pointically by the line from the arbitrary point (i recall that taking 
the arbitrary point as the vertex of a cone through the curve x, this cone, say the 
cone x, meets the torse in the curve x twice, and in the x (r — 2) curve in question) ; 
viz. these points are either points of contact of tangents from the vertex to the 
x (r — 2) curve; or they are double points, or else cusps of the x (1— 2) curve ; in which 
several cases respectively they count 1, 2 or 3 times, among the x(r — 2) (r — 3) points. 
51. The points of contact are the n (x — 2r + 8) points of intersection of the lines 
n with the cone x. We have in fact a plane n through the vertex of the cone, and 
in this plane two consecutive lines of the system; hence at each of the x — 2r + 8 
points the generating line of the cone meets the two consecutive lines of the system; 
that is, there is with the curve x(r— 2) a 2-pointic intersection, not arising out of 
any singularity of the curve, and consequently a contact of this curve with the 
generating line of the cone. 
52. The actual double points of the curve x (r — 2) are first the 2k apparently 
coincident points of the curve x, and secondly the eo (x — 2r +10) points on the lines &>. 
For first if we consider through the vertex a line meeting the curve x in two points, 
say A, B, this meets the torse in these points each twice and in r — 4 other points. 
Now imagine a line from the vertex to the point P in the vicinity of A, this meets 
the torse in the point P twice and in r — 2 points, which are points on the x(r— 2) 
curve; hence as P travels through A, 2 of the r — 2 points come together at B, and 
again separate, that is B is an actual double point on the x (r — 2) curve; and 
similarly A is an actual double point on the curve; and we have thus the 2k double 
points. Secondly, since the line &> is a nodal line on the torse, a generating line of 
the cone, in the neighbourhood of and considered as travelling through one of the 
x — 2r +10 points, meets the torse in two points which come to coincide and then 
again separate; that is each of the x— 2r +10 points is an actual double point on 
the curve x(i—2); and the whole number of these is = eo (x — 2r + 10). 
53. The stationary points of the curve x(r— 2) are first the points on the curve 
m which apparently coincide with the curve x\ viz. the number of these, as was seen, 
is = mx — a — 3/3 — 2y — Sv — 4to — 8H ; secondly, the v (x — 2r + 9) points on the lines v;