A RIGID BODY IN ELLIPTIC SPACE. 
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6. As the triangle of reference take any self-conjugate triangle with respect to the 
Absolute conic, so that the equation to the conic becomes 
px 2 -f- qy' 2 -f- rz 2 = 0. 
Then T^ lz =px- i x^-\-qy^y< 2l -\-rz 1 z % , and 
cos 2 8 = 
{ pxpc 3 + qyyj 2 + rz x z 2 y 
(px{~ 4 - %j\ + rz?)( px* + qyf + rz*) 
This suggests a new system of coordinates. Let (x, y, z) denote the cosines of the 
generalised distances from the angular points of the triangle of reference, of a point 
whose coordinates were {x y , y 1} z x ) in Ordinary Geometry. 
Then 
px i 2 
XX = 
&c. 
Hence 
pxp + qy-f + rz x 2 ’ 
x 2j ry 2 -\-z 2 — 1, 
.and the equation to the Absolute conic is 
£r 3 +y 3 d-2 3 =0. 
Then if 8 denote the distance between two points (x, y, z), {x f , y', z'), 
cos 8 = xxf-\- yy'-\- zz '- 
7. If {l, m, n) be the coordinates of any point, the equation to its polar line with 
respect to the Absolute conic is 
lx-\-my-\-nz— 0. 
Here (l, m, n) may be looked upon as the coordinates of the pole, or the tangential 
coordinates of the line, indifferently ; and we shall always suppose that 
Z 2 +m 3 +n 2 =l. 
The form of the equation shows that a point is distant one right angle from any 
point of its polar. From this theorem, we deduce by Reciprocation, that a given line 
is perpendicular to any line through its pole. Hence the sides and angles of any self¬ 
conjugate triangle with respect to the Absolute conic are all right angles. Such a 
triangle is called a Quadrantal triangle. We can now give a new interpretation of 
the coordinates ( x , y, z); they are the sines of the perpendiculars from the point let 
fall on the three sides of the triangle of reference. 
O 
