132 
ON THE SURFACES THE LOCI OF 
[503 
The Excuboquartic a(3y, 8e, a. 
52. The notion is, that we have a fixed point a, two fixed lines 8, e, and a 
singly infinite series of lines, or say the generating lines of a skew surface: each 
generating line determines, with the point a, a plane; and if in this plane we draw, 
meeting the lines 8, e, a line to meet the generating line in a point P, then the 
locus of this point P is the curve about to be considered. 
53. In the case in question, the singly infinite series of lines is that of the lines 
which meet each of the lines a, /3, y, or say these are the generatrices of the hyper 
boloid a/3<y: the locus, or curve a/3y, 8e, a, is (as mentioned above) an excuboquartic. 
It is not necessary for the purpose of the memoir, but it is interesting to consider in 
conjunction therewith the excuboquartic arising in like manner from the directrices of 
the hyperboloid; it will appear that the two curves are the complete intersection of 
the quadric a/3y by a quartic surface. Observe that the two curves are given as 
follows: viz. considering for the quadric a/3<y any tangent-plane through the point a, 
and drawing in this plane, to meet the lines S and e, a line, this meets the section 
of the quadric surface by the tangent-plane in two points, the locus of which is the 
aggregate of the two curves: viz. the section being a line-pair, the two points belong, 
one of them to a generatrix and the other to a directrix of the quadric surface. 
54. It is convenient to take x = 0, y = 0 for the equations of the line 8 ; z = 0, w = 0 
for those of the line e: for then, for any plane Ax + By + Gz 4- Dw = 0, the line in this 
plane and meeting the lines 8 and e, has for its equations Ax + By = 0, Gz + Diu = 0; 
or, what is the same thing, for the plane P = 0 the equations of the line are 
P xy = 0, P zw = 0, where P xy , P zw denote the terms in x, y and in s, w respectively. 
I take also x 0> y 0> z 0 , w 0 for the coordinates of the point a, and PS — QR = 0 for 
the equation of the quadric surface, P, Q, R, S being given linear functions of (x, y, z, w): 
we have then say P-0R = 0, Q - OS = 0 for the equations of any generatrix, and 
P - (pQ = 0, R - cf)S = 0 for the equations of any directrix of the hyperboloid. 
The equation of the plane through the point a and the generatrix P -OR = 0, 
Q — 68 = 0, is clearly 
(Q o -08 o )(P -OR ) — (P 0 — 0R 0 ) (Q -68 ) = 0; 
so that for the line in this plane, meeting the lines 8 and e, we have 
(Qo- 6So) (P X y — 6Rxy) — (Po ~ ORo) (Qxy — OSxy) = 0, 
(Qo - 0S 0 ) (P zw - 0R ZW ) - (Po - 6Ro) (Qzw - 0S ZW ) = 0 ; 
and joining thereto the equations 
0 P Q Pxy d" P Z w Qxy d~ Qz w 
R 8 Rxy ”b R'ZW Rxy + Rzw 
(equivalent in all to three equations,) the elimination of 0 gives the required curve: 
the equations thus are 
PS — QR = 0, 
(QoS — QS 0 ) (P X yR ~~ PRxy) ~ (PqP ~ PRo) (QxyS — QS xy ) = 0,