PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
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y-tuple point, &c. corresponding- to -the multiple generating lines, if any. A section 
through the directrix line is in general made up of this line counting a times, and of /3 
generating lines through the point which corresponds to the plane of the section ; if 
the section pass also through a y-tuple generating line (y>*/3, in the same way as for 
the scroll S(l, 1, m)), then, of the {3 generating lines, y unite together in the y-tuple 
generating line. The general section through a y-tuple generating line breaks up into 
this line counting y times, and a curve of the order m—y , which has on the directrix 
line an a— y(+|3 — y) tuple point and a cS-tuple point, &c. at its intersections with the 
other multiple generating lines, if any. 
Equation of the Scroll S(l, 1, m) of the Order m , Article Nos. 17 & 18. 
17. Taking for the equations of the directrix lines (#=0, y=0) and (z= 0, w= 0), 
and supposing that these are respectively an a-tuple line and a /3-tuple line on the scroll 
u-\-P=m, it is obvious that the equation of the scroll is 
(*X®, y)\z, ™y= o. 
In fact starting with this equation, if we consider the section by a plane through the 
line (#=0, y= 0), say the plane y='hx, then the equation gives 
i,*)“0, w ) p=0 > 
that is, the section is made up of the line (x=0, y=0) reckoned a times, and of /3 other 
lines in the plane y=7^x; and the like for the section by any plane through the line 
(z= 0, w= 0), say the plane z=vw. Hence the assumed equation represents a scroll of 
the order m, having the two lines for an a-tuple line and a /3-tuple line respectively, 
and conversely such scroll has an equation of the assumed form. 
Case of a y-tuple generating line. 
18. The multiple generating line meets each of the lines (#=0, y= 0) and (z= 0, w=0) ; 
and we may take for the equations of the multiple generating line x-\-y= 0, z-\-w = 0. 
This being so, the foregoing equation of the scroll may be expressed in the form 
(fX>, y) a (z, z+wy*= 0, 
or say 
(u, v, w, ..)(*, z+v>y= o, 
where U, V, W, . . . are functions of the form (*fx, yf. Hence (y$>a or /3), if the func- 
tions U, V, W, ... contain respectively the factors (x-j-y) 7 , (x+y) 7-1 , ( x-\-y) v ~ 2 , . .., the 
equation will be of the form 
z+w) 7 = 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 4-/3 — y), and the scroll will therefore have the line x-\-y=0, z-\-w= 0 as 
a y-tuple generating line. 
MDCCCLXIV. 4 F 
