220 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [340 
where {x, y, z) satisfy the foregoing equation, is the foot of a generating line; and we 
may 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 
X, Y, z, 
w 
' 0B , y , z, 
0 
viz. 
p—Yz—Zy, s —— Wx, 
q = Zx — Xz, t —— Wy, 
r — Xy — Yx, u = — Wz ; 
or, writing for greater convenience — v in the place of IT, 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 + 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 coordi 
nates whereof are {A, B, C, F, G, H); then for the generating lines which meet this 
line we have 
v {Ax + By + Gz) + Fp + Gq + Hr = 0, 
and this equation, together with the equation (*$#, y, z) n = 0, determines (x, y, z), the 
coordinates of the foot of a generating line which meets the arbitrary line {A, B, G, F, G, H). 
Since the order of the scroll is = n, the number of such generating lines should be 
= n, that is, there should be n relevant intersections of the two curves, 
v {Ax + By + Gz) + Fp + Gq + Hr = 0, 
(*]$>» y> z ) n = o; 
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, <7 = 0, r = 0, v = 0 
have in common 6 intersections, and let these be points of the multiplicities a 1; a 2 , a 3 , ... a d 
on the curve (* ][x, y, 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 x = 0, 1, 2, &c., as the case may be, and so for the 
other intersections) ; then the kn points of intersection include the + a 2 .... + a 0 , or 
say the 2a intersections; but these, being independent of the line {A, B, G, F, G, H) 
under consideration, are irrelevant points, and the number of relevant points of inter 
section is kn — 2a ; that is, if we have l t a = {k — l)n, then the scroll in question, viz. 
the scroll generated by a line which meets the plane w = 0 in the curve {*Qx, y, z) n = 0, 
and which has for its six coordinates {p, q, r, vx, vy, vz), will be a scroll of the ?ith 
order.