76] 
445 
76. 
ON THE TRIPLE TANGENT PLANES OF SURFACES OF THE 
THIRD ORDER. 
[From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 118—132.] 
A surface of the third order contains in general a certain number of straight 
lines. Any plane through one of these lines intersects the surface in the line and in 
a conic, that is in a curve or system of the third order having two double points. 
Such a plane is therefore a double tangent plane of the surface, the double points (or 
points where the line and conic intersect) being the points of contact. By properly 
determining the plane, the conic will reduce itself to a pair of straight lines. Here 
the plane intersects the surface in three straight lines, that is in a curve or system of 
the third order having three double points, and the plane is therefore a triple tangent 
plane, the three double points or points of intersection of the lines taken two and 
two together being the points of contact. The number of lines and triple tangent 
planes is determined by means of a theorem very easily demonstrated, viz. that through 
each line there may be drawn five (and only five) triple tangent planes. Thus, 
considering any triple tangent plane, through each of the three lines in this plane 
there may be drawn (in addition to the plane in question) four triple tangent planes: 
these twelve new planes give rise to twenty-four new lines upon the surface, making 
up with the former three lines, twenty-seven lines upon the surface. It is clear that 
there can be no lines upon the surface besides these twenty-seven ; for since the three 
lines upon the triple tangent plane are the complete intersection of this plane with 
the surface, every other line upon the surface must meet the triple tangent plane in 
a point upon one of the three lines, and must therefore lie in a plane passing through 
one of these lines, such plane (since it meets the surface in two lines and therefore 
in a third line) being obviously a triple tangent plane. Hence the whole number of 
lines upon the surface is twenty-seven; and it immediately follows that the number 
of triple tangent planes is forty-five. The number of lines upon the surface may also 
be obtained by the following method, which has the advantage of not assuming