560 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
m, n, p are each of them situate on the same scroll 2, the curve m meeting each gene- 
rating line of 2 in a points, the curve n each generating line in (3 points, and the curve 
p each generating line in y points. Each generating line of % is a(3y times a generating 
line of S, and we have S=S‘ fSy S', where S' may be a proper scroll ; it is however to be 
noticed that if the curves to, n, p any two of them intersect, S' will itself break up and 
contain certain cone-factors, as will presently appear. And if X, instead of being a 
proper scroll, be a torse, then we may consider S as degenerating into S', the reduction 
in order being of course =a(3y X order of £. 
3. But this is not the only way in which the scroll S (to, n, p) may degenerate ; for sup- 
pose that two of the directrix curves, say n and p, intersect, then the lines from the 
point of intersection to the curve to form a cone of the order to which will present 
itself as a factor of S ; and generally if the curves n and p intersect in a points, the 
curves p and to in (3 points, and the curves to and n in y points, then we have a cones 
each of the order to, (3 cones each of the order n, and y cones each of the order p, or 
say S=CS', where C is the aggregate of the cone-factors; and the scroll S degenerates 
into S', the reduction in order being =am-\-(3n-{-yp. It is hardly necessary to remark 
that if a point of intersection of two of the curves is a multiple point on either or each 
of the curves, it is, in reckoning the number of intersections of the two curves, to be 
taken account of according to its multiplicity in the ordinary manner. 
4. There is yet another case to be considered : suppose that the curves n and p lie on 
a cone, and that the curve to passes through the vertex of this cone ; this cone, repeated 
a certain number of times, is part of the locus, or we have S=C"S', so that the scroll S 
degenerates into S', the reduction in order being =dx order of cone. If, to fix the ideas, 
the curves n and are respectively the complete intersections of the cone by two sur- 
faces of the orders g , h respectively (this implies n=gk, p=hk, if k be the order of the 
cone), which surfaces do not pass through the vertex of the cone, and if, moreover, the 
vertex of the cone be an a- tuple point on the curve to, then 6=.agh, and the reduction 
in order is —aglik. 
5. The foregoing causes of reduction, or some of them, may exist simultaneously ; it 
would require a further examination to see whether the aggregate reduction is in all 
cases the sum of the separate reductions. But the aggregate reduction once ascertained, 
then writing S(to, n, p) for the order of the reduced scroll, we shall have 
S(to, n, p)=2mnp— Reduction. 
In particular, in the case above referred to, where the curves n and p, p and to, to and n 
meet in a, (3, y points respectively, but there is no other cause of reduction, 
S(to, n , p)= 2mnp — am —(3n—yp, 
which is a formula which will be made use of. 
The foregoing investigations apply, mutatis mutandis , to the scrolls S(m 2 , n), S (to 3 ) ; 
but I do not at present enter into the development of them in regard to these scrolls. 
