180 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
The 8 formulae, particular cases, Articles 36 to 40. 
36. The S formulae have in fact been obtained in the investigation of the G 
formulae: we have 
S (to, n, p) = 2mnp, 
S (to 2 , n) = n ([to] 2 + M), 
8 (to 3 ) = [to] 3 + M (to — 2). 
37. In confirmation of the formula 8(1, to 2 ) = [to] 2 + M, it is to be remarked that 
if we take through the line 1 an arbitrary plane, this meets the curve to in to 
points, and joining these two and two together we have \ [to] 2 lines, each of them 
meeting the curve to twice and also meeting the line 1; that is, the lines in 
question are generating lines of the scroll 8(1, to 2 ). The line 1 is, as already 
mentioned, an (h = ) (^ [to] 2 + M)tuple line on the scroll; the section by the arbitrary 
plane is therefore the line 1 taken (•£ [to] 2 + M) times, together with the before-mentioned 
\ [to] 2 lines; that is, the order of the surface is [to] 2 + M, as it should be. This is 
in fact the mode in which the order of the scroll >8(1, to 2 ) was originally obtained 
by Dr Salmon. 
38. As regards the formula 8 (to 3 ) = £ [to] 3 -+- M (to — 2), suppose that the curve to 
is a (p, q) curve on the hyperboloid, we have as before m=p + q, M — —pq, and the 
formula becomes 
8 (to 3 ) = ^ [p + q] 3 — pq (p + q — 2), 
which is 
=ib] 3 +l№ 
viz. as already remarked, the surface is in this case the hyperboloid taken £ [p] 3 + ¡t [g] 3 
times. 
39. It is to be noticed also that if the curve to be a system of lines (to — m,M= 0), 
then the formula gives 
8 (to 3 ) = ^ [to] 3 , 
which is right, since in this case the scroll is made up of the ^ [to] 3 hyperboloids 
generated each of them by a line which meets three out of the to lines. 
In the case of a curve to, which is such that the coordinates of any point of 
the curve are proportional to rational and integral functions of the order to of an 
arbitrary parameter 6, or say the case of a unicursal curve of the order to, we have 
(A = 1[to-1] 2 and :.)M = — (to — 1), 
and the formula gives 
S (to 3 ) = t [to — l] 3 , 
for a direct investigation of which see post, Annex No. 1.