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A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [340 
construction, and we can only say that the scroll S given by the construction is the 
a ggregate of the scrolls S', S",and the like when we have the scrolls S', S",..., 
each repeated any number of times, or say when S — S' a S"P... Suppose however that 
the scrolls S', S",.. are any one or more of them a torse or torses—or, to make at 
once the most general supposition, say that we have S = 2$', where 2 is a torse, or 
aggregate of torses (2 = 2 /a 2^...), and S' is a proper scroll or aggregate of proper 
scrolls; then, although it is not obligatory to do so, we may without impropriety throw 
aside the torse-factor 2, and consider the original scroll S as degenerating into the 
scroll S', and as suffering a reduction in order accordingly. 
2. As an illustration, consider the scroll S (to, n, p) generated by a line which 
meets three directrix curves of the orders to, n, p respectively; and assume that the 
curves to, n, p are each of them situate on the same scroll 2, the curve on meeting 
each generating line of 2 in a points, the curve n each generating line in ft points, 
and the curve p each generating line in y points. Each generating line of 2 is afty 
times a generating line of S, and we have S = 2 ai *yS', 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 cone-factors, as will presently appear. And 
if 2, instead of being a proper scroll, be a torse, then we may consider S as degene 
rating into S', the reduction in order being of course = afty x order of 2. 
3. But this is not the only way in which the scroll S (to, n, p) may degenerate; 
for suppose 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 on 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 ft points, and the curves on and n in 7 points, then 
we have a cones each of the order to, ft cones each of the order 01, and 7 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 + ftn + 7p. 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 d S', 
so that the scroll S degenerates into S', the reduction in order being = 6 x order of 
cone. If, to fix the ideas, the curves 01 and p are respectively the complete inter 
sections of the cone by two surfaces of the orders g, h respectively (this implies 
n — gk, p = Kk, 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 = aghk. 
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