A RIGID BODY IN ELLIPTIC SPACE. 
289 
The form of this equation shows that a circle is a conic having double contact with 
the Absolute in the two points in which it is met by the polar line of the centre of 
the circle. 
Many other interesting properties of conics may be worked out by means of these 
equations, but'as they will not concern us we pass on at once to the geometry of three 
dimensions. 
Solid geometry. 
13. As before, we refer the Absolute quadric to a self-conjugate tetrahedron. Let 
the equation to it, in any system of plane coordinates of ordinary space, be 
pod-{-qft 2 -\-ry 2 -f .sS 3 = 0. 
Then, if 6 be the distance between the points 1, 2, 
Pnc , 2/9—_ [+ gfllAi + rr /^2 + s8 Af _ 
[i }a i + c ifti + r Yi 2 + s ^i 2 ][i ja 2 2 +i'Aj 2 ++sSd] 
This again suggests a new system of homogeneous coordinates. Let {x, y, z, u) 
denote the cosines of the distances of the point (a, /3, y, §) from the four angular 
points of the tetrahedron of reference. Then 
_ V* - ■_ &c 
~px 2 + q/3* + rry* + s&’ ' 
For any real point 
i q i q i o t 
%~ry J r z -\-u— 1 . 
The equation of the Absolute quadric in these coordinates is 
» 2 +r+2 2 -HF=o. 
Also if 6 be the distance between two points (x, y, z, u) (x, y, z, u), we have 
cos d—XX '+yy'+ zz -\-uu ’. 
14. If {l, m, n, p) be the coordinates of a point the equation of the polar plane with 
reference to the Absolute quadric is 
lx-\- my -\-nz-\- \m — 0. 
In the equation to any plane we shall suppose the coefficients such that 
l~ ffi TO 3 + n 2 -\-p ' 1 
2 r 
MDCCCLXXXIV. 
