208 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [340 
where U, V, W,... are functions of the form (*$#, y) a . Hence (7 a or /3), if the 
functions U, V, W,... contain respectively the factors (x + y) y , (x + y)y~ 1 , {x + y) y ~ 2 ,..., the 
equation will be of the form 
( *1[x + y, z + w) y — 0 
(the coefficients being functions of x, y, z and z + w, or, what is the same thing, x, y, z, w, 
of the order a + /3 — 7), and the scroll will therefore have the line x + y = 0, z +w = 0 as 
a 7-tuple generating line. 
Equation of the Scroll S( 1, 1, m) of the order m, Article Nos. 19 to 24. 
19. We may take # = 0, y = 0 for the equations of the twofold directrix line, 
z = 0 for the equation of the plane of the curve m (an arbitrary plane section of the 
scroll). Then (a + /3 = m), if the curve m have at the point {x=0, y = 0), or foot of 
the directrix line, an a. (+ /3)tuple point, and if moreover we have y = 0 for the equation 
of the common tangent of the ¡3 branches (viz. if the plane y — 0, instead of being 
an arbitrary plane through the directrix line, be the plane through this line and the 
common tangent of the (3 branches), the equation of the curve m will be of the form 
2 {ywf (*$0, y) a+p 23 = 0, 
where the summation extends to all integer values of /3' from 0 to /3, both inclusive. 
20. Taking y = \x for the equation of any plane through the directrix line, then 
the corresponding point on the directrix line will be the intersection of this line 
(x = 0, y = 0) by the plane £ = 6w, where 0 = + ; the foot of the directrix line is 
ch ~f~ ct 
given by the value 0 = 0, or \ = —, and the equation of the line of approach is 
Cf 
h 
therefore y = x; this should coincide with the line y = 0, which is the common 
G/ 
tangent of the /3 branches; that is, we must have b = 0; I retain, however, for the 
moment the general value of b. 
21. The equations of a generating line will be 
y = \x, z= 0w — px ; 
and then taking X, Y, (Z = 0) and W for the coordinates of the point of intersection 
with the curve m, we have 
Y = \X, O = 0W-pX, 
X(FTTr(*][X, Yy + f>-* = 0, 
and thence 
= 0; 
or, what is the same thing, 
20-0' (Xpf (*£1, x) a+ 0- 2 0'