TRANSACTIONS OF SECTION A. 647 
I adopt the form given them by Professor Killing in his well-known work on 
‘Nicht-Euclidische Raumformen.’ 
The fundamental equations are 
OL a ene 
da ain y’ Ee cos y; ¢ f(b) da’ 
where a, 6, ¢ are the sides of a triangle, a, 8, y their angles, a opposite a, 8 opposite 
6, &c. The solution of these equations is 
(a) £% TAO Ki . 
sna sin8 siny 
from which, by appropriate partial differentiations and eliminations, we obtain 
[Aap _ CAMP __ [feP___e 
T-(f@P 1-(7OFP I-LhOF 
where «? is a constant. A final integration now determines the form of the 
function f(a) ; it is 
F(a) =i ltt is aes : 
where C is the constant of integration. 
By defining 
sin a= ane —e~a*) 
(read sine of a with respect to the modulus x) the equation for f (a) is more con 
cisely written 
J(@) =sin,, (@+C). 
It is easy to show that /(0)=0, so that C isa period, that is, a multiple of 
«r/ —1, and therefore 
SJ(a@) =sin, (@ + nikr) = + sin, a. 
We choose the positive sign in this last equation, assigning arbitrarily, for the 
proper relative directions of the sides of our triangle, and equation (a) now 
becomes 
sin,@_sin,d_ sin,¢ 
sna sinB siny 
These are the fundamental equations of trigonometry. Out of these the entire 
theory of measurement proceeds in the usual way. 
The theory of measurement thus constituted is purely ideal. There is no real 
universe that can be measured by it without the arbitrary assumption of a definite 
value for x, and there are only three kinds of value for x possible. These are 
kK’ =00, «*=a positive real number, x?=a negative real number. What has 
appeared very startling to the modern world is that there is as yet no theory of 
knowledge that can tell us which of these three diverging paths we must take. 
This is an old story told in a new way. 
8. Report on Recent Progress in the Problem of Three Bodies. 
By EK. T. Wuirraker, Jf.A. 
See Reports, p. 121. 
9. On Singular Solutions of Ordinary Differential Equations. 
By Professor A. R. Forsytu, Sc.D., LBS. 
