PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
577 
imagine this generating line determined by means of the coordinates (X, Y, Z, W), given 
functions of (x, y, z) of a point on the line. This being so, the “ six coordinates,” say 
(p, q, r, s, t, u), of the line are 
I X, Y, Z, W 
I x, y, z, 0 
VIZ. 
p=Yz — Zy , s=—Wx, 
q—Zx—YLz, t = —Wy, 
r=X.y—Yx, u = —Wz; 
or, writing for greater convenience — v in the place of W, the six coordinates of the line 
are p, q, r, vx, vy, vz, where p, q, r are functions of (x, y, z), connected by the relation 
px-\-qy-j-vz= 0 ; and v is also a function of (x, y, z). 
52. Consider the intersection of the surface by an arbitrary line the six coordinates 
whereof are (A, B, C, F, G, H); then for the generating lines which meet this line we 
have 
?;( A# + By + Gs) + F^ + G# + Hr = 0 . 
And this equation, together with the equation y, z) n = 0, determines (x, y, z ), the coor- 
dinates of the foot of a generating line which meets the arbitrary line (A, B, C, F, G, H)- 
Since the order of the scroll is equal n, the number of such generating lines should be 
=n , that is, there should be n relevant intersections of the two curves, 
v( Aa?-f- By -f C z) + Fp + Gq + Hr = 0, 
(*Xr, y, z) n = 0. 
But if ( p , q, r, vx, vy, vz) are each of the order k, the number of actual intersections 
is =kn, which is too many by \k— l)n. 
53. Suppose that the curves 
p— 0, q= 0, r=0, vx=0, vy= 0, vz= 0, 
or say the curves 
p=0, q— 0 , r=0, r=0 
have in common 6 intersections, and let these be points of the multiplicities a„ a 2 , ot 3 ...ct 9 
on the curve V-> z ) n — 0 (viz. according as the curve does not pass through any one 
of the intersections in question, or passes once, twice, &c. through such intersection, we 
have for that intersection a, = 0, 1, 2, &c., as the case may be, and so for the other inter- 
sections); then the kn points of intersection include the a ,+ a 2 +a fl , or say the %<z, 
intersections ; but these, being independent of the line (A, B, C, F, G, H) under considera- 
tion, are irrelevant points, and the number of relevant points of intersection is kn—Xu; 
that is, if we have Xu=(k — 1 )n, then the scroll in question, viz. the scroll generated by 
a line which meets the plane w=0 in the curve (*$x, y, z) n =0, and which has for its 
six coordinates (p, q, r, vx, vy, vz), will be a scroll of the nth. order. 
4 H 
MDCCCLXIV. 
