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XVI. A Second Memoir on Skew Surfaces, otherwise Scrolls. By A. Cayley, F.B.S. 
Received April 29 , — Read May 26, 1864. 
The principal object of the present memoir is to establish the different kinds of skew 
surfaces of the fourth order, or Quartic Scrolls ; but, as preliminary thereto, there are 
some general researches connected with those in my former memoir “ On Skew Surfaces, 
otherwise Scrolls”*, and I also reproduce the theory (which may be considered as a 
known one) of cubic scrolls ; there are also some concluding remarks which relate to 
the general theory. As regards quartic scrolls, I remark that M. Chasles, in a footnote 
to his paper, “ Description des courbes de tous les ordres situees sur 'les surfaces reglees 
du troisieme et du quatrieme ordres ” f, states, “ les surfaces reglees du quatrieme ordre 
.... admettent quatorze especes.” This does not agree with my results, since I find only 
eight species of quartic scrolls ; the developable surface or “ torse ” is perhaps included 
as a “ surface reglee but as there is only one species of quartic torse, the deficiency is 
not to be thus accounted for. My enumeration appears to me complete, but it is possi- 
ble that there are subforms which M. Chasles has reckoned as distinct species. 
On the Degeneracy of a Scroll , Article Nos. 1 to 5. 
1. A scroll considered as arising from any geometrical construction, for instance one 
of the scrolls S(m, n,jp), S(m 2 , n), S(m 3 ) considered in my former memoir, or say in 
general the scroll S, may break up into two or more inferior scrolls S', S", . .; but as long 
as S', S", . . are proper scrolls (not torses, and a fortiori not cones or planes), no one of 
these can be considered, apart from the others, as the result of the geometrical construc- 
tion, and we can only say that the scroll S given by the construction is the aggregate 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'“S"A . . 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 sup- 
position, say that we have S = 2S', where 2 is a torse, or aggregate of torses (2=2'“2" /5 . . .), 
and S' is a proper scroll or aggregate of proper scrolls ; then, although it is not obliga- 
tory 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(m, n, q>) generated by a line which meets 
three directrix curves of the orders m, n, jp respectively ; and assume that the curves 
* Philosophical Transactions, vol. cliii. (1863), pp. 453-483. 
f Comptes Rendus, t. liii. (1861), see p. 888. 
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