44 
SECTION A. 
and by adding the two series together we get 
a = 1 -f- Aa 2 4 
A 2 « 
21 
+ 
A 3 a 
— e Aa ~ 
Also 
f .A —A Aa A 
(—a) = a z=e = e 
— A ' a? 
and 
2 tr—A A a 
a = e 
% 
So far as angle is concerned, irrespective of the whole amount of turn 
ing, we have 
a —A _ a?*-A' 
n_ TT_ 
It follows that Aa 2 is the logarithm of aA ; and a? the logarithm of a 1 . 
As the most general expression for minus is a ( 2 ”+ 1 )’ r > 
log (—1) = (2?i-|-l)7r * aF. 
The general expression for -j/—i is a^^ nn ’ therefore 
7T 7T 
Zoÿ 4/—1 = (2n7r4-f )• 0 s ; and for 4- it is d?‘ niT , therefore log 4- = Inn • a 2 . 
TV 
Hence generally log (aa A ) = Zop a 4- -4 • a 2 ". 
In his Geometric de Position Carnot says, in reference to the celebrated 
discussion about the logarithms of negative quantities “Quoique cette 
discussion soit aujourd’hui terminé, il reste ce paradoxe savoir que quoiqu’ 
on ait log (—z) 2 = log O) 2 , on n’a cependant pas 2 log (—z) = 2 log z.” 
The paradox may be explained as follows : Suppose the complete ex 
pression for z to be j?a 2w,r , then that for —z is za( 2n + 1 ) ,r ; then 
71* 7T 
log z 2 =2 log z 4- 4«?r • a 2 " &x\dlog (—z) 2 — 2logz-{- (4n+2) tt • a 2 . 
As the latter is twice the logarithm of za( 2n + 1 ) ,r , the supposed paradox 
vanishes. 
To prove that 
a A ft B =' cos A cos B— sin A sin B cos a[3 
tr ir 7r 
4- cos A sin B • 15 2 4- cos Bsin A’ a? — sin A sin B sin aft • aft 2 - 
Since a A = cos A 4- sin A • a?, 
and ft B = cos B 4- sin B • ft 
by multiplying the two equations together we obtain 
7T 7T 7T 7T 
ftB — cos ^ cos 43 4. cc>s ^4 gfra H ‘ ft 2 4~ cos B sin A • d2 4- sin A sin B • a^/5 2 ". 
Now, as was shown in the previous paper (p. 98) 
a- ft 2 = — cos aft — sin aft • aft^ -