T 
[339 
9), G (m 2 , n-), 
ly as regards 
for ND (m 3 )) 
hrough three 
point ; that 
four points ; 
mts through 
curve there 
oil whatever, 
its (e.g. the 
tion for the 
asses a line 
' points, or 
1 any point 
rn be drawn 
the assumed 
ts in which 
generating 
Lree points); 
joints. The 
3 times, or 
croll S'(m 3 ), 
le line; the 
scroll S (m*) 
every point 
nes through 
ines through 
through p 
r if p, q, ... 
pass a. lines 
tgh q points 
lines each 
ON SKEW SUlt F ACES, OTHERWISE SCROLLS. 
175 
through three points and counting as a single line. It is to be added that, in the 
case just referred to, the a lines will generate a scroll S'(m 3 ) taken [pP times, the 
/3 lines will generate a scroll S" (to 3 ) taken l [gp times, &c., which scrolls together 
make up the scroll S (m 3 ), or say 
S (m 3 ) = £ [pp. S'(m 3 ) + [gp. S" (m 3 ) 4- &c.; 
it may however happen that, e.g. of the a lines, any set or sets or even each line will 
generate a distinct scroll or scrolls—that is, that the scroll S'(m 3 ) will itself break up 
into scrolls of inferior orders. 
20. A good illustration is afforded by taking for the curve m a curve on the 
hyperboloid or quadric scroll 1 ; such curves divide themselves into species; viz. we have 
say the (p, q) curve on the hyperboloid, a curve of the order p + q meeting each 
generating line of the one kind in p points, and each generating line of the other 
kind in q points; here 
in = p + q, (h = P [p] 2 +1 [gp, and .'.) M = - pq. 
Assuming for the moment that p, q are each of them not less than 3, it is clear 
that the lines through three points which can be drawn through any point of the 
curve are the generating line of the one kind counting as p [p — lp lines through 
three points, and the generating line of the other kind counting as \ \_q — lp lines 
through three points, so that 
L [mp + m — 2 + M = i [p — lp -f \ [q — lp. 
The complete scroll S(m 3 ) is made up of the hyperboloid considered as generated by 
the generating lines of the one kind taken l [pp times, and the hyperboloid con 
sidered as generated by the generating lines of the other kind taken £ [gp times (so 
that there is in this case the speciality that the surfaces S'(m 3 ), S" (m 3 ) are in fact 
the same surface). And hence we have 
S (m 3 ) = (2 (i [pp +1 (pp) =) } [pp + l [qj. 
21. I notice also the case of a system of m lines. Taking here a point on one 
of the lines, the (h — m 4- 2 =) [mp — m + 2 lines through three points which can be 
drawn through this point are the ^ [m — lp lines which can be drawn meeting 
a pair of the other (m — 1) lines, and besides this the line itself counting as one line 
through three points (^ [m — lp + 1 = \ [ni\ 3 — m + 2) ; the line itself, thus counting as a 
single line through three points, is not to be reckoned as a line through four or 
more points drawn through the point in question, that is, the system is not to be 
regarded as a curve through every point of which there passes a line through four 
points: each of the lines is nevertheless to be counted as a single line through four 
points, and (since there are besides two lines which may be drawn meeting each four 
of the m lines) the total number of lines through four points is = Pj [mp + m. 
22. In the following investigations for G (m, n, p, q), &c., the foregoing special 
cases are excluded from consideration; it may however be right to notice how it is 
1 It is hardly necessary to remark that {reality being disregarded) any quadric surface whatever is a 
hyperboloid or quadric scroll.