84] 
TWO SURFACES OF THE SECOND ORDER, 
491 
which is therefore the equation of the Intersect-Developable for systems of the case 
in question. The equation may also be presented under the form 
{q (px? + rz s ) (liy 2 — gz 2 +• 2pzw) —p (qy 2 + rz~) (— hx 2 + fz 2 + 2qzw)) 2 
+ 4p 2 q 3 x 2 y 2 (hy 2 — gz 2 + 2pzw) (— hx 2 + fz 2 + 2qzw) — 0, . 
which it is to be remarked contains the extraneous factor z 2 . The following is also 
a form of the same equation, 
[r (qg —pf)z 3 —fp 2 zx 2 + gq 2 zy 2 + 2pq (px 2 + qy 2 ) w} 2 
— 4pq (px 2 + qy 2 + rz 2 ) {r (hy 2 — gz 2 ) (fz 2 — hx 2 ) + 2pq (gx 2 — fy 2 ) zw} = 0. 
C. Two surfaces forming a system belonging to this class may be represented by 
equations of the form 
a x 2 + b y 2 + 2/ yz + 2n zw = 0, 
a'x 2 + b'y 2 + 2f'yz + 2 n'zw = 0, 
in which bn' — b'n = 0, af' — af= 0. In these equations x = 0 is the equation of a 
properly chosen plane passing through the two conjugate points, y = 0 is the equation 
of the single conjugate plane, z = 0 that of the triple conjugate plane, and w — 0 is 
the equation of a properly chosen plane passing through the single conjugate point. 
Or without loss of generality, we may write 
a (x 2 — 2yz) + /3 (y 2 — 2zw) = 0, 
a! (x 2 — 2yz) + /3' (y 2 — 2zw) — 0, 
where x, y, z and w have the same signification as before 1 . The result after all reduc 
tions is 
4fZ 3 w 2 4- 12y 2 z 2 w + 9y*z — 24}X 2 yzw — 4tx 2 y 3 + 8x*w = 0, 
which may also be presented under the forms 
z (y 2 — 2zw) 2 — 4y (y 2 — 2zw) (x? — 2yz) + 8w (x 2 — 2yz) 2 = 0, 
and z (3y 2 + 2zw) 2 — \x 2 (y 3 — 2x 2 w + Qyzw) = 0. 
[In these three equations and in the last two equations of p. 495 as originally 
printed, there was by mistake, an interchange of the letters x and y.] 
1 Of course in working out the equation of the Intersect-Developable, it is simpler to employ the equations 
x 2 -2yz = 0, y 2 -2ziv — 0. These equations belong to two cones which pass through the Intersect and have their 
vertices in the triple conjugate point and single conjugate point respectively. I have not alluded to these cones 
in the text, as the theory of them does not come within the plan of the present memoir, the immediate 
object of which is to exhibit the equations of certain developable surfaces—but these cones are convenient 
in the present case as furnishing the easiest means of defining the planes x = 0, w — 0. If we represent for 
a moment the single conjugate point by S and the triple conjugate point by T (and the cones through these 
points by the same letters), then the point T is a point upon the cone S, and the triple conjugate plane 
which touches the cone S along the line TS touches the cone T along some generating line TM. Let the 
other tangent plane through the line TS to the cone T be TM', where M' may represent the point where 
the generating line in question meets the cone S; and we may consider M as the point of intersection of 
the line TM with the tangent plane through the line SM’ to the cone S: then the plane TMM' is the 
plane represented by the equation # = 0, and the plane SMM* is that represented by the equation w = 0. We 
may add that y = 0 is the equation of the plane TSM', and z — () that of the plane TSM. 
62—2