120 PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
cc, (3, y, $ being linear functions of the current coordinates (X, Y, Z, W), viz. we have 
«=( . , H, -G, A£X, y, Z, W), 
i3=(-H, ,, F,BX „ ), 
y=( G, -F, • , CJ „ ), 
&=(— A, — B, — C, . X „ )• 
72. It thus appears that when AF+BG+CH is not =0, the reciprocal of the 
scroll 
(H, F, C, B, A-F, -GXjp, & rf = 0 
has an equation of the very same form, 
(H, F, C, B, A-F, -GXp, q, r) 2 =0 ; . . . . (Rec. I.) 
so that in fact the scroll, tenth species, S(3 2 ), defined as the scroll generated by a line 
in involution which passes through two points of a skew cubic, may be reciprocally defined 
as the scroll generated by a line in involution which lies in two osculating planes of a 
skew cubic. 
73. If for (a, (3, y, b) we substitute their values in terms of (X, Y, Z, W), the fore- 
going equation of the reciprocal scroll is obtained as an equation of the fourth order in 
the coordinates (X, Y, Z, W), and (in the first instance) of the fifth degree in the coeffi- 
cients (A, B, C, F, G, H). It is a remarkable circumstance that the whole equation 
contains the constant factor AF+BG+CH, so that throwing this out, the reduced 
equation will be only of the third degree in the coefficients. 
74. The transformation is a very troublesome one, but I will indicate the steps by 
which I succeeded in accomplishing it. Each of the functions (p, q, r) is a quadric 
function of (X, Y, Z, W), say, 
we have to form the value of 
(H, F, C, B, A-F, -GXp, q, r) 2 , 
viz. representing this for shortness by 
(H, F, C, B, A-F, -GX 
the coefficient of X 4 is 
that of X 3 Y is 
