ON SKEW SURFACES, OTHERWISE SCROLLS. 
[From the Philosophical Transactions of the Royal Society of London, vol. сын. (for 
the year 1863) pp. 453—483. Received February 3,—Read March 5, 1863.] 
It may be convenient to mention at the outset that, in the paper “ On the 
Theory of Skew Surfaces ” ( x ), I pointed out that upon any skew surface of the order n 
there is a singular (or nodal) curve meeting each generating line in (n — 2) points, 
and that the class of the circumscribed cone (or, what is the same thing, the class 
of the surface) is equal to the order n of the surface. In the paper “On a Class 
of Ruled Surfaces ”( 1 2 ), Dr Salmon considered the surface generated by a line which 
meets three curves of the orders m, n, p respectively: such surface is there shown to 
be of the order = 2mnp; and it is noticed that there are upon it a certain number 
of double right lines (nodal generators); to determine the number of these, it was 
necessary to consider the skew surface generated by a line meeting a given right line 
and a given curve of the order m twice; and the order of such surface is found to 
be l) + h, where h is the number of apparent double points of the curve. 
The theory is somewhat further developed in Dr Salmon’s memoir “ On the Degree 
of a Surface reciprocal to a given one”( 3 ), where certain minor limits are given for the 
orders of the nodal curves on the skew surface generated by a line meeting a given 
right line and two curves of the orders m and n respectively, and on that generated 
by a line meeting a given right line and a curve of the order m twice. And in 
the same memoir the author considers the skew surface generated by a line the 
equations whereof are (a, . . \t, l) m = 0, (a', . . \t, l) n = 0, where a, . . a', . . are any linear 
functions of the coordinates, and t is an arbitrary parameter. And the same theories 
are reproduced in the “ Treatise on the Analytic Geometry of Three Dimensions ”( 4 ). 
1 Cambridge and Dublin Math. Journ. vol. vii. pp. 171—173 (1852), [108]. 
2 Ibid. vol. viii. pp. 45, 46 (1853). 
3 Trans. Royal Irish Acad. vol. xxiii. pp. 461—488 (read 1855). 
4 Dublin, 1862. [Ed. 4, 1882.]