312 
[410 
410. 
A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clix. (for 
the year 1869), pp. Ill—126. Received May 30,—Read June 18, 1868.] 
The present Memoir is supplementary to my “ Second Memoir on Skew Surfaces, 
otherwise Scrolls,” Phil. Trans, vol. cliv. (1864), pp. 559—577, [340], and relates also to 
the theory of skew surfaces of the fourth order, or quartic scrolls. It was pointed out to 
me by Herr Schwarz^), in a letter dated Halle, June 1, 1867, that in the enumeration 
contained in my Second Memoir I have given only a particular case of the quartic 
scrolls which have a directrix skew cubic; viz. my eighth species, $(1, 3 2 ), where 
there is also a directrix line. And this led me to observe that I had in like 
manner mentioned only a particular case of the quartic scrolls with a triple directrix 
line; viz. my third species, S (1 3 , 1, 4), where there is also a simple directrix line. 
The omitted species, say, ninth species, S (1 3 ), with a triple directrix line, and tenth 
species, S (8 2 ), with a directrix shew cubic, are considered in the present Memoir; and 
in reference to them I develope a theory of the reciprocal relations of these scrolls, 
which has some very interesting analytical features. 
The paragraphs of the present Memoir are numbered consecutively with those of 
my Second Memoir above referred to. 
Quartic Scroll, Ninth Species, S (1 3 ), with a triple directrix line. 
54. Consider a line the intersection of two planes, and let the equation of the 
one plane contain in the order 3, that of the second plane contain linearly, a variable 
parameter 0; the equations of the two planes may be taken to be 
(p, q, r, s\0, 1) 3 = 0, (u, F£0, 1) = 0, 
1 I take the opportunity of referring to his paper on Quintic Scrolls, Schwarz, “ Ueber die geradlinigen 
Flachen fiinften Grades,” Crelle, t. lxvii. (1867), pp. 23—57.