SECTION III, 1916 _ (171) : Trans. RS.C. 
Alternate Numbers as Geometrical Indices. 
by J.C: GLASHAN, LED} E-R.S.C: 
(Read May Meeting, 1916). 
In the following paper, symbols printed in Clarendon type denote 
geometrical points, lines, angles and figures. 
1. If A and B, two points on the endless line 1, divide 1 into seg- 
ments (1) and )l(, and if P be a point on (1), then AB will denote 1 
considered as described through A, P and B in the order A, P, B; 
and BA will denote the same line considered as described through the 
same three points but in reverse order, viz., B, P, A. Also (AB) will 
denote the segment (1) considered as described from A through P to 
B; and (BA) will denote the same segment considered as described 
from B through P to A. If P and Q are two points on the segment 
(AB), the segments (AP) and (QB) are said to be condirectional and 
the lines AP and QB to be coincident; also the segments (AP) and 
(BQ) are said to be contradirectional and the lines AP and BQ to be 
contraincident. If R is a point on AB so taken that (AB) and (BR) 
are condirectional, R is said to lie on (AB) produced. The segment 
‘(AB) produced’ may be denoted by A) (B+. 
The straight line is an endless line with one given point 
on it, the point at infinity. Since the point at infinity must lie either 
on (1) or on )I(, provided it is not one of the determining points of 
these segments, its location will serve to distinguish between the 
two segments. If it lie on )1(, the points on (1) are said to lie between 
the determining points of (1). If A and B cut the straight line 1 into 
the segments (AB) and A) (B, it will hereafter be taken for granted 
unless otherwise specifically stated, that the point at infinity lies on 
A) (B, cutting that segment into two segments A) (B+ and, + A) (B. 
2. If the lines a and d intersect, the point of intersection will be 
denoted by ad; if the lines AB and CD intersect, the point of inter- 
section will be denoted by AB. CD. The endless line through ab and 
cd will be denoted by ab | cd, and the segment from ab to cd by 
(ab | cd). Ifthe points A, B and C are collinear, their line of collin- 
ation will be denoted by A|BJ|C. If the lines a, b and ¢ are con- 
current, their common point will be denoted by a. b.c. 
3. If the straight lines AB and CD intersect at O, the angle 
described by the radian OR turning counter-clockwise on the pole 
