11 
647] 
THE EQUATION, SYMMETRICAL DETERMINANT = 0. 
55 
this is loo : m : n = a' : b! : viz. if g=g oX + g J y ) h = h 0 x + h^y, this is 
lx : m : n = a : g 0 + g x 6 : h 0 + h,6, 
or joining hereto the equation y = 9x, we have 
lx : ly : m : n = a : ad : g a +g 1 0 : h 0 + hrf, 
where l, a, g 0 , g 1} h 0 , /q are constants; m, n are linear functions of the coordinates 
{ x > 2/» w )- The equations represent, it is clear, a right line which is the polar in 
question; and they may be written 
lx _ /qm - #1% ly _ h 0 m—g 0 n 
a Kgo-Kg^ a Kg^-h^' 
When the plane passes through a ray, the conic becomes, as was stated, the point- 
pair composed of the two nodes in such ray; the harmonic in regard to these two. 
points of the intersection of the ray with the nodal line is thus a point on the 
polar: that is, the polar meets the ray; and the two nodes are situate harmonically 
in regard to the intersections of the ray with the nodal line and the polar respectively. 
The polar may be arrived at in a different manner, viz. if instead of a plane 
through the nodal line we consider a point on the nodal line, this is the vertex of 
a circumscribed quadric cone; and taking the polar plane of the nodal line in regard 
to this cone, then considering the point as variable, the different polar planes all pass 
through a line which is the polar in question. And hence, taking for the point the 
intersection of the nodal line with an axis, it appears that the axis meets the polar; 
and, moreover, that the two tropes through the axis are harmonics in regard to the 
planes through the axis, and the polar and nodal line respectively. 
Collecting the foregoing results, we have a quartic surface as follows: 
We have two lines, a nodal line and a polar; meeting each of these, four lines 
called “rays” and four other lines called “axes.” On each ray, harmonically in regard 
to its intersections with the nodal line and the polar, two nodes of the surface (in 
all 8 nodes): through each axis, harmonically in regard to the planes through it and 
the nodal line and the axis respectively, two tropes of the surface (in all 8 tropes). 
In each trope (or, what is the same thing, in its conic of contact) are 4 nodes; 
through each node (or, what is the same thing, touching its tangential quadricone) are 
4 tropes; the relation of the nodes and tropes may be thus represented, viz. taking 
the pairs of nodes to be 1, 2; 8, 4; 5, 6; 7, 8; and those of tropes to be I, II; 
III, IV; V, VI; VII, VIII; then we have