The collected mathematical papers of Arthur Cayley.

410]

A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS.

325

or, what is the same thing,
- 6 (3X - Yy + Z3 - 3 Wa,) (a22 - 6a/Sy + + 4a + 43 - 3/372) = 0, (Rec. III.)
as may be verified by actual substitution of the values of the coordinates.
78. By what precedes, substituting for p, q, r their values in terms of a,,/3,,,
it appears that we have the remarkable identity
(H, F, C, B, A- F, -G/38 -y2, 3-, ay - 2)2
= (AF+ BG OH CH) x X( (- a2 + 3aS37 - 2/3 )
+ Y (- a/3 + 2ary2 - 82y)
+Z ( a7y-2/28 + 3yZ 2)
l+ W( a82 — 3/y8 $+273 )j
79. In the case above considered of the tenth species, S(32), for which
AF + BG +CH not= 0, the three forms of the reciprocal equation are of course absolutely
equivalent to each other. The first form has the advantage of putting in evidence
the fact that the reciprocal scroll is also of the tenth species; the other two forms
do not, at least obviously, put in evidence any special property of the reciprocal scroll.
Reciprocals of Eighth Species, S(1, 32), and Ninth Species, S(13).
80. If AF+BG + CH =0, then the equation
(H, F, C, B, A- F, - Gp, q, r)2= 0
is a scroll of the eighth species, S(1, 32). The first form of the reciprocal equation
becomes identically 0=0, on account of the evanescent factor AF + BG + CH, but the
second and third forms continue to subsist, and either of them may be taken as the
equation of the reciprocal scroll. Taking the third form, and calling to mind the
significations of (a, /3, 7, 8), viz.
a =(., H, -, A X, Y, Z, W),
S=(- H, F.,., B, ),
7=( G, -F,,, ),
8=(-A, -B, -C,.,, ),
it is to be observed that a = 0, 3= 0, y =0, = 0 are the equations of four planes
passing through a common line, viz. the line whose coordinates are (A, B, 0, F, G, H),
and the equation thus puts in evidence that this line is a triple line on the reciprocal