88 
ME. A. CATLET’S SIXTH jMEMOIE ETOX QUAXTICS. 
and finally, the expression 
reducing the arc to its sine 
for the distance of the point x, y, z from the line t). 
and omitting the evanescent factor, is 
X , 
z 
9.^ 
r 
q^t 
To 
s / 2 ) • 
223. If in the above formula we put (^, q, 
as usual then the line-equation of the Absolute is f + ??"=0, or what is the 
same thing, the Absolute consists of the two points in which the line ^=0 mtersects the 
line-pair ; the last-mentioned line-pair, as passing through the Absolute, is by 
definition a circle ; it is in fact the circle radius zero, or an evanescent cii-cle. If we 
put also the coordinate ^ equal to unity, then the preceding assumption as to Hie coor- 
dinates of the points of the Absolute must be understood to mean only a- ; I : ? : t • 
oj. 1 ; —i ; 0; that is, we must have x and y infinite, and, as before, x —0, oi in 
other words, the Absolute will consist of the points of intersection of the line infinityby 
the evanescent circle x^H-/=0. With the values in question, 
224. The expression for the distance of the points {x, y) and {x\ ij') is 
^{x~-xJ-V{y~-y'f; 
that for the distance of the lines (|, n, 1) and (|', rj, f) is 
0r+w _ 
_ . iv-ry 
“Sin + 
which may also be written 
=tan“'^ — tan~*^ 5 
and the expression for the distance of the point (x, y) from the line (|', >? , 0 
which are obviously the formulae of ordinary plane geometry, {x, y) bemg ordinary 
rectangular coordinates, 
225. The general formulae suffer no essential modification, but they are greatly sim- 
plified in form by taking for the point-equation of the xkbsolute, 
or what is the same, for the line-equation 
f4-;,*+zr'=0. 
In fact, we then have for the expression of the distance of the points {x, y, z\ {x , y,z). 
ssx' + Ijy’ +22^ 
