[339 
340] 
201 
intersection 
which line is 
rse (to, n) in 
by considering 
oectively. The 
intersection of 
ct as follows: 
tangent to n, 
rating line of 
ng lines which 
intersection of 
meet the line 
m the line of 
ts in which 
therefore in 
line of inter- 
ve to from the 
line meets the 
the curve to ; 
are v tangents 
is = n/M + mv, 
ated by a line 
ag ¡л, it is to 
‘ Geometry of 
have 
desirable ; but 
nemoir, I have 
most readily 
340. 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE 
SCROLLS. 
[From the Philosophical Transactions of the Royal Society of London, vol. cliv. (for the 
year 1864), pp. 559—576. 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’’(A 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 foot 
note to his paper, “Description des courbes de tous les ordres situées sur les surfaces 
réglées du troisième et du quatrième ordres”( 2 ), states, “les surfaces réglées du quatrième 
ordre.... admettent quatorze espèces.” 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 réglée ; ” but as there is only one species of quartic torse, 
the deficiency is not to be thus accounted for. My enumeration appears to me com 
plete, but it is possible 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 (to, n, p), S (to 2 , n), S (to 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 à fortiori not cones or planes), 
no one of these can be considered, apart from the others, as the result of the geometrical 
1 Philosophical Transactions, vol. сын. (1863), pp. 453—483, [339]. 
2 Comptes Bendas, t. lui. (1861), see p. 888. 
C. Y. 26