590 
A SIXTH MEMOIR UPON QUANTICS. 
[158 
223. If in the above formula we put (p, q, r) = (l, i, 0), (p 0 , q 0) r 0 ) = ( 1, —i, 0), 
where as usual i = V — 1, then the line-equation of the Absolute is + y 2 = 0, or what 
is the same thing, the Absolute consists of the two points in which the line z = 0 
intersects the line-pair x 2 + y 2 = 0 ; 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 
circle. If we put also the coordinate z equal to unity, then the preceding assumption 
as to the coordinates of the points of the Absolute must be understood to mean only 
x : y : 1 = 1 : i : 0, or 1 : — i : 0 ; that is, we must have x and y infinite, and, as 
before, x 2 + y 2 = 0, or in other words, the Absolute will consist of the points of inter 
section of the line infinity by the evanescent circle x 2 + y 2 — 0. With the values in 
question, 
224. The expression for the distance of the points (x, y) and (x', y) is 
V (x - xj + (y- yj ; 
that for the distance of the lines (£, y, Ç) and (£', y, £') is 
+ w' 
-1 »» '/I 
V£ 2 + y 2 V£' 2 + y 2 
which may also be written 
p 
= tan -1 - — tan -1 
= tan -1 - 
V 
and the expression for the distance of the point (x, y) from the line (£', y, ^') is 
Ç'x + y'y + F 
V + y' 2 
which are obviously the formulae of ordinary plane geometry, (x, y) being ordinary 
rectangular coordinates. 
225. The general formulae suffer no essential modification, but they are greatly 
simplified in form by taking for the point-equation of the Absolute 
x 2 + y 2 + z 2 = 0, 
or, what is the same, for the line-equation 
f+ 7f+ £ 2 = 0. 
In fact, we then have for the expression of the distance of the points (x, y, z), (x', y' z'), 
cos -1 
xx' + yÿ + zz'