MATHEMATICS AND ASTRONOMY. 
49 
e A COS W B COS W g A sin to • a 2 + B sin W • |3 2 - 
Because e" 4 cos w and e B G0S w are independent of axis, they can be changed 
from the order in which they occur in the sum of indices. 
The meaning of a A ft B is the sector of the spiral which joins the begin 
ning of the former with the end of the latter. 
Hence when ¡3 = a, 
Tr 
„A B A A 4- B) cos w . {A + B) sin w • a 2 
a w a w — e e 
w 
==e (A+B)a 
== a A + B 
w 
which is the addition theorem for the logarithmic spiral, the two compo 
nent sectors being in the same plane. 
Exponent of a compound angle. 
We have 
e *aW = 1+ * a ^S 5 + ^-(aV)®+ii («W+; 
where a A 13 B is expanded as shown above, and is double of the 
compound angle, (a^/9 8 ) 3 is three times the compound angle and so on. 
It is to be observed that (a' 4 /? -8 ) 2 is not in general equal to a ¡3 • 
Let * = A = B = | and let /3 be identical with a, then we have 
IT 7T 
?I a 
= 1 
But e^ a =e 2 and it is also = a 2 “ ; 
7T 
__7T 7T 
and thus e 2 = a 2 “ » 
ivhich is a rational expression for the celebrated equation of Euler 
l/—i * / ~ 1 — e~ 2 • 
By taking logs we obtain 
hat is 
a. 2 log {a 2 ) = — TS' 
log (a“) 
— •?* 
Vo differentiate a A . 
Aa 
= cos A-\-sin A' a 2 '* 
Since 
2