70 
[314 
314. 
ON THE CURVES SITUATE ON A SURFACE OF THE SECOND 
ORDER. 
[From the Philosophical Magazine, vol. xxn. (1861), pp. 35—38.] 
A surface of the second order has on it a double system of generating lines, 
real or imaginary; and any two generating lines of the first kind form with any two 
generating lines of the second kind a skew quadrangle. If the equations of the 
planes containing respectively the first and second, second and third, third and fourth, 
fourth and first sides of the quadrangle are x = 0, y — 0, z = 0, w = 0, and if the 
constant multipliers which are implicitly contained in x, y, z, w respectively are suit 
ably determined, then the equation of the surface of the second order (or say for 
shortness the quadric surface) is xw — yz = 0. 
Assume 
V = P 
x X ’ 
z 
X 
V ,, p v 
-, then -, or 
P x P 
say (X, p, v, p), may be regarded as the co 
ordinates of a point on the quadric surface; we in fact have x : y : z : w = 1 , 
X p Xp 
or what is the same thing, =Xp : /xp : vX : /xv. The four quantities (X, p, v, p) are 
for symmetry of notation used as coordinates; but it is to be throughout borne in 
mind that the absolute magnitudes of X and p, and of v and p are essentially 
indeterminate; it is only the ratios X : p and v : p that we are concerned with. 
An equation of the form 
(*$>> APU. p) q = o, 
that is, an equation homogeneous of the degree p as regards (A, p), and homogeneous 
of the degree q as regards (v, p), represents a curve on the quadric surface; and this 
curve is of the order p + q. In fact, combining with the equation of the curve the 
equation of an arbitrary plane 
Ax + By + Cz + Die = 0,