338 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
[411 
29. For the off-point or singularity 6 = 1; this is a point on the cuspidal curve 
at which the second derived functions all of them vanish. In further explanation hereof 
consider a surface U= 0, and the second polar of an arbitrary point (a, /3, 7, 8) ; viz. 
this is (ad x + /3d y + yd z + 8d w ) 3 JJ = 0, or say for shortness A 2 /7=0, where the coefficients 
of the powers and products of (a, /3, 7, 3) are of course the second derived functions of U; 
this equation, when reduced by means of the equations of the cuspidal curve, may acquire 
a factor A, thus assuming the form A (aP 4- /3Q + 7R + 8S) 3 = 0, and if so the intersections 
of the cuspidal curve with the second polar (= 2a + 0, if, as for simplicity is supposed, 
there is no nodal curve) will be made up of the intersections of the cuspidal curve 
with the surface A = 0, and of those with the surface aP + f3Q + 7R + 8S = 0 each 
twice ; the latter of these, depending on the coordinates (a, /3, 7, 8) of the arbitrary 
points, are the points cr each twice; the former of them, or intersections of the cuspidal 
curve with the surface A = 0, are the points 6, or off-points of the cuspidal curve. 
If there is a nodal curve, the only difference is that the off-points are such of the 
above points as do not lie on the nodal curve. 
30. As the most simple instance of the manner in which this singularity may 
present itself, consider a surface FP 2 + GQ 3 = 0, where the degrees of the functions are 
f, p, g, g, and therefore n =/+ 2p = g 4- Sq, if n be the order of the surface. This has 
a cuspidal curve P = 0, Q = 0 of the order pq\ the equation A' 2 (FP' 2 + GQ 3 ) = 0 of the 
second polar, when reduced by the equations P = 0, Q = 0 of the cuspidal curve, becomes 
simply F (AP) 2 = 0 ; and we have thus the off-points F = 0, P = 0, Q = 0, consequently 
0=fpq. 
31. But suppose, as before, the case of a surface (A, B, C][P, Q) 3 = 0 having a 
cuspidal curve P = 0, Q = 0, and therefore AC — B 1 being =0 for P = 0, Q = 0. The 
equation of the second polar, writing therein P = 0, Q- 0, becomes (A, B, C][AP, AQ) 2 = 0, 
and if for any given surface this assumes the form A (i¥AP + JSfAQy = 0 (observe that 
M, JS r may be fractional provided only the MAP + NAQ is integral), then there will 
be on the cuspidal curve the off-points A = 0, P = 0, Q = 0. 
32. An interesting example is afforded by a surface which presents itself in the 
Memoir on Cubic Surfaces: the surface 
43/8 
— 4<y 3 x (x- -f 3zw) 
+ zw (3P + zw) 3 = 0 
has the cuspidal conic y = 0, 3P + zw = 0, and (as coming under the form FP 2 + GQ 3 = 0) 
has the off-points zw = 0, y= 0, 3P + zw = 0 ; that is, the points (x = 0, y — 0, z = 0), 
[x = 0, y = 0, w — 0) each twice ; 6 = 4. 
But writing the same equation in the form 
(4, Qx, 8x 2 + zw][y 3 — 2x 3 , x 3 — zw) 3 = 0, 
4 . (8P + zw) — (6#) 2 = — 4 (x 3 — zw), 
where