410] 
A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
313 
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, q, r, 8§y, -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 
(*$«> VY + 2 (*$>> y) 3 + w (*'$>, y) s = 0 ; 
and we may, by changing the values of z and w, make the term in (x, y) 4 to be 
(*\x, y) 4 + (ax + fry) (*$«, y) 3 + (yx + 8y)(*'\x, y) 3 , 
where the arbitrary constants a, /3, y, 8 may be so determined as to reduce this to a 
monomial kx A , ka?y, or kxhf. 
56. The coefficient k may vanish, and the equation of the scroll then is 
Z (*\v, y) 3 + W (*'\x, y) 3 = 0, 
or, what is the same thing, it is 
(*$#, y) 3 (z, w) = 0, 
viz. the scroll has in this particular case the simple directrix line z = 0, w = 0, thus 
reducing itself to the third species, S (1 3 , 1, 4), with a triple directrix line and a single 
directrix line. It is proper to exclude this, and consider the ninth species, $(1 3 ), as 
having a triple directrix line, but no simple directrix line. 
57. The scroll S( 1 3 ) may be considered as a scroll S(m, n, p) generated by a 
line which meets each of three given directrices; viz. these may be taken to be the 
directrix line, and any two plane sections of the scroll. The section by any plane is 
a quartic curve having a triple point at the intersection with the directrix line; 
moreover the sections by any two planes meet in four points, the intersections of the 
scroll by the line of intersection of the two planes. Conversely, taking any line and 
two quartics related as above (that is, each quartic has a triple point at its inter 
section with the line, and the two quartics meet in four points lying in a line), the 
lines which meet the three curves generate a quartic scroll S (1 3 ). This appears from 
the formula 
S(m, n, p) = 2mnp — am — (3n — yp (Second Memoir, No. 5); 
we have in the present case 
m — 1, n = 4, p = 4<, a = 4, (3 — 3, y = 3, 
and the order of the scroll is 32-4-12-12, =4, that is, the scroll is a quartic 
scroll; there is no difficulty in seeing that through each point of the line there pass 
40 
C. VI.