284 
MR. R. S. HEATH ON THE DYNAMICS OF 
r 
where c is an arbitrary constant. Hence, in this generalised system of Geometry, the 
distance between two points on a line is measured by the logarithm of the anharmonic 
ratio of the range formed by the two points and the absolute point-pair of the line, 
multiplied by an arbitrary constant. 
4. The relations between lines passing through a point and lying in a plane are 
exactly the same as the relations between points along a line. Among the lines lying 
in a plane and passing through a point there are two fixed lines called the Absolute 
pair of lines; these are the pair of tangents that can be drawn from the point to the 
Absolute conic of the plane. The measurement of angles will thus be exactly similar 
to the measurement of distances. The angle between two lines lying in a plane is 
measured by the logarithm of the anharmonic ratio of the pencil formed by the lines 
and the Absolute pair of lines passing through the point, multiplied by an arbitrary 
constant. There is a special advantage in choosing both these arbitrary constants 
to be -> where i denotes \ —1. 
From these definitions it follows by properties of poles and polars that the distance 
between two points is equal to the angle between their polars, so that any theorem of 
distances has a reciprocal theorem relating to angles. 
5. Let U = 0 be the equation to the Absolute conic in any plane in the notation of 
Ordinary Geometry. If (aq, y x , z x ), (x 2 , y 2 , z 2 ) be any two points, the coordinates of 
any point on the line joining them are proportional to aq— Aaq, y 1 — \y. 2 , z 1 —A z, 2 . 
Hence to find the Absolute point-pair we have the equation 
Tj n — A U 2 2—0 .... 
with the usual notation. 
Let 8 be the generalised distance between the points 1, 2. Then 
h 
« 
where \ 1? \ 2 are the roots of the quadratic equation (1). Hence 
therefore 
that is, 
g28z_j_ g—28;- 
y+y 
A^ A^ 
e~'-t-e 5 'j ~_ (Aj+Aj ) 2 
~ 4A 1 A 2 
9 
COS 2 8: 
U 12 
IT U 
L n-' 22 
( 2 ). 
