PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
113 
Quartic Scroll , Tenth Species, (3 2 ), with a directrix skew cubic met twice by each 
generating line *. 
59. Consider a line the intersection of two planes; and let the equation of each 
plane contain in the order 2 a variable parameter d ; the equations of the two planes 
may be taken to be 
(p, q, rjO, 1) 2 =0, (p', q', r'X#, 1) 2 =0, 
where ( p , q, r, p, q', r') are linear functions of the coordinates [x, y, z, w) ; hence eli- 
minating $, we have as the equation of the scroll generated by the line in question, 
□ =0, where □ is the resultant of the two quadric functions. The equation may he 
written 
4 (pq' — p'q ) (rq'— r'q ) — ( pr' —p'rf = 0 ; 
and the scroll has thus as a nodal (double) line the skew cubic determined by the 
equations 
" p, q, r 
p\ i, r 1 
1=0. 
It is easy to see (and indeed it will be shown presently) that this curve is met twice by 
each generating line of the scroll, and that the scroll is consequently a quartic scroll as 
described above. 
60. The coordinates ( x , y, z, w) may be fixed in such manner that the equations of 
the skew cubic shall be 
or, what is the same thing, 
x, V> z 
y, z , w 
yw—z 2 =0, zy—xw= 0, xz—y 2 = 0; 
each of the equations pql —p'q=0, rq'—r'q—0,pr'—p'r=0 is then the equation of a 
quadric surface passing through the skew cubic, or, what is the same thing, each of the 
functions pq' — p'q, r<l — r'q, pr 1 — p'r is a linear function of yw — z 2 , zy — xw, xz — y 2 ; and 
the equation of the scroll is given as a quadric equation in the last-mentioned quantities. 
It will be convenient to represent the equation in the form 
(H, F, C, B, A-F, -GXyw-z 2 , zy—xw, xz-y 2 ) 2 ^ 0, 
or, writing for shortness 
yw—z 2 , zy—xw, xz—y 2 —p, q, r, 
which letters ( p , q, r) are used henceforward in this signification only, the equation 
* I Have worded this heading in accordance with that of the eighth species, Second Memoir, No. 47, but the 
two headings might be expressed more completely thus : 
Eighth Species, S(l, 3jj), with a directrix line and a double directrix slcew cubic met twice by each generating line ; 
Tenth Species, S(3|), with a double directrix shew cubic met twice by each generating line ; 
viz. the subscript 2 would indicate that the skew cubic is a nodal (double) line on the scroll, the exponent 2 
indicating that it is met twice by each generating line. 
Q2 
