of Edinburgh , Session 1879 - 80 . 655 
on the same side as before by travelling once more through a length 
L. 
This carious conclusion is an immediate result of the fact that 
straight lines are re-entrant and intersect only once. (In double 
elliptical space the apparent anomaly does not occur on account of 
the double intersection.) 
The best way of representing the thing to the mind that I can 
think of is to imagine a rigid body composed of three rectangular 
arrows Ix, I y, I z (fig. 14). Ix slides along OA; I y passes through 
a ring which slides on OB (being long enough never to slip out) ; 
I z is, of course, determined in position when Ix and I y are fixed in 
any positions. 
In starting from 0, ,let Ix and I y be horizontal and Iz vertical ; 
then slide Ix along OA. Ix will at last return along A'O. The ring 
will return along B'O. It is obvious, therefore, that, at our first 
return to 0, Iz must be downwards, for, since the system of arrows is 
rigid, one who plants himself with feet at I, head at z and looks along 
Ix must see y to his left as he did at starting. 
It is obvious that during the journey I y as well as Iz has rotated 
through 180°, a repetition of the process rotates both through 180° 
more, and then everything is as before. 
If we cause a complete straight line of length L to revolve through 
360°, always remaining perpendicular to a given line, it will sweep 
out the two sides of a complete plane. 
It follows at once, therefore, that the area of a complete plane, 
taking into account both sides, is finite, and the same for every 
complete plane. This I shall call P in the meantime. We also 
see, in accordance with what was proved before, that the two sides 
of the complete plane are not distinct, since we can pass continuously 
upon the plane from a point on one side to the same point on the 
other side. 
Those who find difficulty in realising this property of the plane in 
single elliptic space should take a ribbon of paper, twist it through 
180°, and then gum the ends together. A surface is thus formed 
which has the property that one can trace a continuous line upon it 
from a point on one side to a point exactly opposite on the other side. 
After what has been laid down the following propositions are 
obvious. 
4 i 
VOL. X. 
