of Edinburgh, Session 1879 - 80 . 661 
This is the equation for Elliptic Space ; that for Hyperbolic Space is of course 
d ? D 
dr 2 
From the equation (2) we get at once 
(2 ')* 
D = pa sin - , (3) 
P 
r being measured from the intersection of the lines, and the constants of inte- 
gration determined by the condition 
D — ar 
when r is infinitely small compared with p, which of course includes the con- 
dition D = 0 when r — 0. 
The corresponding formidse for Hyperbolic Space are 
= pa sin h — (4) 
P 
for a pair of intersectors ; and 
r r 
D = d( e p + e p 
= d cos h L . . . (5) 
P 
for a pair of non-intersectors, r being measured in the one case from the inter- 
section, in the other from the points of minimum distance. 
From the formulae (3), (4), and (5) all the trigonometry of Elliptic and 
Hyperbolic Space can be deduced most readily. I append one or two applica- 
tions, and select for my purpose important formulae, but anything like a 
complete development would be out of place here, t 
* The differential equations (2) and (2') contain all the metrical properties of elliptic and 
hyperbolic space. (2) suggests that a pair of straight lines diverging at a small angle from a 
point might intersect again in distinct points any number of times. The proposition proved 
above for elliptic space generally, that all the lines radiating from any point intersect in the 
same second point, seems, however, to compel us to conclude that at the point where any line 
intersects another for the second time, it must return into itself ; for a line can be brought by 
continuous rotation into coincidence with its prolongation, hence we must reach the same 
second point of intersection in whichever direction we proceed from the first point. I can 
see no way out of this at present ; and if there is none, it would appear that we cannot get 
beyond double elliptic space, even if we can consistently get so far. 
f I may refer the reader to Frischauf, “ Elemente der Absolute Geometrie,” Leipzig, 1876 ; 
Lobatschewsky, Crelle, xvii. p. 295 ; Klein, Annalen der Mathematik, iv. p. 573, vi. p. 112, &c. ; 
Cayley, Annalen der Mathematik, v, p. 630. 
