200 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
be understood a plane passing through a tangent line of the curve. The intersection 
of two consecutive tangent planes is a line meeting the two curves, which line is 
the generating line of the Torse, and such Torse is therefore the Torse (m, n) in 
question. 
The foregoing investigation is not very satisfactory, but I confirm it by considering 
the case of two plane curves, orders m and n, and classes ¡x and v, respectively. The 
tangents of the two curves can, it is clear, only meet on the line of intersection of 
the planes of the curves; and the construction of the Torse is in fact as follows: 
from any point of the line of intersection draw a tangent to m and a tangent to n, 
then the line joining the points of contact of these tangents is a generating line of 
the Torse. The order of the Torse is equal to the number of generating lines which 
meet an arbitrary line; and taking for the arbitrary line the line of intersection of 
the two planes, it is easy to see that the only generating lines which meet the line 
of intersection are those for which one of the points of contact lies on the line of 
intersection; that is, they are the generating lines derived from the points in which 
the line of intersection meets one or other of the two curves; they are therefore in 
fact the tangents drawn to the curve n from the points in which the line of inter 
section meets the curve m, together with the tangents drawn to the curve m from the 
points in which the line of intersection meets the curve n. Now the line meets the 
curve n in n points, and from each of these there are /a tangents to the curve m; 
and it meets the curve m in m points, and from each of these there are v tangents 
to the curve n; hence the entire number of the tangents in question is = n\x + mv, 
which confirms the theorem. 
Annex No. 5.—Order of Torse (m 2 ) (referred to, Art. 46). 
We have here to find the order of the developable or Torse generated by a line 
meeting a curve of the order m twice, viz., the class of the curve being /a, it is to 
be shown that we have 
Torse (m 2 ) = (m — 3)/x. 
I deduce the expression from the formula given p. 424 of Dr Salmon’s ‘ Geometry of 
Three Dimensions; ’ viz. putting in his formula /3 = 0, and /a for his r, we have 
Order = 4) — | a = m/A — (4m + 4a), 
where (see p. 234 et seq.) 
and thence 
so that we have 
fx = m(m — 1) — 2 h, 
= (n — m) = 3 m (m — 2) — 6h — m, 
3/a — ^a = 4m, or 4m + \ a. = 3/a, 
Order = (m — 3) /¿. 
A more complete discussion of the Torses (m, n) and (m 2 ) is obviously desirable; but 
as they are only incidentally connected with the subject of the present memoir, I have 
contented myself with obtaining the required results in the way which most readily 
presented itself.