584 
A SIXTH MEMOIR UPON QUANTICS. 
[158 
P', P'" are a circle having P" for its centre; and so on: and again in the opposite 
direction, a point P v such that P\ P' are a circle having P for its centre; a point 
P" such that P, P" are a circle having P' for its centre, and so on. We have a 
series of points... P", P', P, P', P", ... at equal intervals of distance: and if we take 
the points P, P' indefinitely near to each other, then the entire line will be divided 
into a series of equal infinitesimal elements; the number of these elements included 
between any two points measures the distance of the two points. It is clear that, 
according to the definition, if P, P', P" be any three points taken in order, then 
Dist. (P, P’) + Dist. (.F, P") = Dist. (P, P"), 
which agrees with the ordinary notion of distance. 
211. To show how the foregoing definition leads to an analytical expression for 
the distance of two points in terms of their coordinates, take 
(a, b, cjoc, yf = 0 
for the equation of the Absolute. The equation of a circle having the point (xy') 
for its centre is 
{a, b, c$x, y) 2 {a, b, c$V, yj cos 2 6 - {(a, b, c$>, y\x', y')) 2 = 0 ; 
and consequently if (x, y), (x", y") are the two points of the circle, then 
Q, b, c\x, yffV, y') (a, b, c$y, y'\x", y") 
V(a, b, c$x, y) 2 V(a, b, c$Y, y') 2 V(a, b, c$af, y') 2 V(a, b, c#x", y"Y 
an equation which expresses that the points (x", y") and (x, y) are equidistant from 
the point (x\ y). It is clear that the distance of the points (x, y) and (x', y') must 
be a function of 
b, c\x, y\x\ y') 
V(a, b, c\x, y) 2 V (a, b, c\x\ y') 2 ’ 
and the form of the function is determined from the before-mentioned property, viz. 
if P, P', P" be any three points taken in order, then 
Dist. (P, P') + Dist. (P\ P") = Dist. (P, P"). 
This leads to the conclusion that the distance of the points (x, y), (x', y') is equal to 
a multiple of the arc having for its cosine the last-mentioned expression (see ante, 
No. 168); and we may in general assume that the distance is equal to the arc in 
question, viz. that the distance is 
cos“ 1 (a, b, c\x, y\x\ y') 
V(a, b, c\x, y) 2 V(a, b, c$V, y') 2 ’ 
or, what is the same thing, 
sirr 
(ac — b 2 ) (xy — x'y) 
V(a, b, c\x, y f V (a, b, c\x', yj ‘