172 Proceedings of Royal Society of Edinburgh. [sess. 
Note that the expression SUg + YUg is not the complete 
equivalent of U^j the binomial is a reduced equivalent. For, 
if f3 is variable, the result of differentiating e W-ip will be different 
from the result of differentiating cos b + sin b (J - 1/3). 
If we enquire for the analogous expression for a hyperbolic 
angle, we find that there is none furnished by Algebra. It is 
not e 6 , for 
6 n 7 b 2 b* 
e - 1 +6 + 2 j + 3 | + 
and there is here no ground for breaking up the series into two 
components ; all the terms are real, and so add directly. For the 
same reason it cannot be e~ b . But we know that 
cosh b 
, b 2 ¥ 
1 + 2l + n + ’ 
sinh & = & + 
b b 
5! + ; 
there must therefore be some proper way of expressing a hyperbolic 
angle by means of an exponential function. Try the effect of 
dropping J - 1 from the circular expression eV-i£. We get 
(b/3) 2 (b/3) 3 
+ + ^ +. 
Now introduce the corresponding principle of reduction, namely, 
(3 2 = + 1 ; then 
A = 1 + b P + ^ + Y\ fS+ 
+ ( b+ l + w +)l3 
=sw+W 
if q' denotes a hyperbolic quaternion. Hence it appears that e b P 
is the proper expression for the angle of an equilateral hyperbola. 
It follows that the expression for the spherical quaternion is 
re W- which, after expansion and reduction, gives the spherical 
complex quantity of the form x + yj _ \/3. Similarly the ex- 
pression for the equilateral hyperbolic quaternion is re b P, which, 
after expansion and reduction, gives the hyperbolic complex 
quantity of the form x + y/3- In the former case we have 
r — Jx 2 + y 2 ) in latter, r = Jx 2 - y 2 . Suppose the objection 
