446 
ON THE TRIPLE TANGENT PLANES OF 
[76 
a priori the existence of a line upon the surface. Imagine the cone having for its 
vertex a given point not upon the surface and circumscribed about the surface, every 
double tangent plane of the cone is also a double tangent plane of the surface, and 
therefore intersects the surface in a straight line (and a conic). And, conversely, if 
there be any line upon the surface, the plane through this line and the vertex of the 
cone will be a double tangent plane of the cone. Hence the number of double tangent 
planes of the cones is precisely that of the lines upon the surface. By the theorems 
in Mr [Dr] Salmon’s paper “ On the degree of a surface reciprocal to a given one,” 
Journal, vol. H. [1847] p. 65, the cone is of the sixth order and has no double lines 
and six cuspidal lines: hence by the formula in Pliicker’s “ Theorie der algebraischen 
Curven,” [1839] p. 211, stated so as to apply to cones instead of plane curves, viz. n 
being the order, x the number of double lines, у that of the cuspidal lines, и that of 
the double tangent planes, then 
и = \n(yi — 2) (n 2 — 9) — (2x + 3y) (n 2 — ?г — 6) + 2x (x — 1) + бжу + fy (y — 1), 
the number of double tangent planes is twenty-seven, which is therefore also the 
number of lines upon the surface. 
Suppose the equation of one of the triple tangent planes to be w = 0, and let 
x = 0, у = 0, be the equation of any two triple tangent planes intersecting the plane 
w = 0 in two of the lines in which it meets the surface. Let z = 0 be the equation 
of a triple tangent plane meeting w— 0 in the remaining line in which it intersects 
the surface. The equation of the surface of the third order is in every case of the 
form wP + lexyz = 0, P being a function of the second order, but of the four different 
planes which the equation z = 0 may be supposed to represent, one of them such 
that the function P resolves itself into the product of a pair of factors, and for the 
remaining three this resolution into factors does not take place. This will be obvious 
from the sequel: at present I shall suppose that the plane z = 0 is of the latter class, 
or that P = 0 represents a proper surface of the second order. Since x = 0, у = 0, z = 0, 
are treble tangent planes of the surface, each of these planes must be a tangent 
plane of the surface of the second order P = 0, and this will be the case if we assume 
p = X 2 + y 2 + £ 2 + w 2 
+ yz (mn + ¿¿) + ■г* («* + ;g) + ■(*» + 4) + (l + [) + yw (m + i) + ZW (n + i) ; 
and considering x, y, z and w as each of them implicitly containing an arbitrary 
constant, this is the most general function which satisfies the conditions in question. 
We are thus led to the equation of the surface of the third order: 
U — w |ж 2 + у 2 + z* + w 2 -t- 
yz ( mn + { nl+ s) + xy { lm +¿)+ xw ( г + 7) + yw ( m + й) + zw (”+SI +kxyz = °-