II. A Third Memoir on Skew Surfaces , otherwise Scrolls. By Professor Cayley, F.B.S. 
Received May 30, — Read June 18, 1868. 
The present Memoir is supplementary to my “ Second Memoir on Skew Surfaces, other- 
wise Scrolls,” Phil. Trans, vol. cliv. (1864), pp. 559-577, 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, S(l, 3 2 ), where there is also a direc- 
trix line. And this led me to observe that I had in like manner mentioned only a par- 
ticular case of the quartic scrolls with a triple directrix line ; viz. my third species, 
S(l 3 , 1, 4), where there is also a simple directrix line. The omitted species, say, ninth 
species , S(l 3 ), with a triple directrix line , and tenth species , S(3 2 ), with a directrix skew 
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(l 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 para- 
meter Q ; the equations of the two planes may be taken to be 
(p, q, r, sX$, 1 ) 3 =0, (u, vfd, 1) = 0, 
where (p, q, r, s, u, v ) are any linear functions whatever of the coordinates (x, y, z, w). 
Hence eliminating 6 we have as the equation of the scroll generated by the line in 
question 
(p 2, r, sfv, —u) 3 =0, 
viz. this is a quartic scroll having the line u= 0, v=0 for a triple line; that is, the line 
in question is a triple directrix line. 
55. Taking x=0, y— 0 for the equations of the directrix line, or writing u—x , v=y, 
and moreover expressing ( p , q, r, s) as linear functions of the coordinates (x, y, z, w), 
the equation of the scroll takes the form 
(*X#> #) 4 +z(* 30, 3/) 3 +w(*'X^ 30 3 =0 ; 
* I take the opportunity of referring to his paper on Qnintic Scrolls, Schavarz, “ Ueber die geradlinigen 
Flachen fiinften Grades,” Crelle, t. lxvii. (1867), pp. 23-57. 
MDCCCLXIX. Q 
