OF ALGEBRAIC QUANTITIES. 
343 
But g is inclined to unity at an angle = Q + q c, 
g is inclined to unity at an angle = m . a + p c ; 
.'. g is in length = and is inclined to unity at an angle = m . a + p c. 
18. Let a be a positive quantity, and let g be the m th possible power of a ; 
o 
then g will also be the m* general power of a. 
0 
For since a is a positive quantity, and g the m th possible power of a , (by 
0 
Treatise, Art. 65.) g is a positive quantity ; 
Also, from the nature of possible logarithms, since g is the m th possible power 
of a ; possible hyperbolic logarithm of g = m X possible hyperbolic logarithm 
0 
of a , 
that is (by Art. 9.) g' = m a!, 
0 o 
.-. (by Art. 15.) g is the m ^ general power of a. 
0 
1 9. Let a be any quantity whatever, and let g be the m th possible power of 
a, then g will also be the general power of a. 
i> v 
For let & be a positive quantity, in length = a, 
and let a be inclined to unity at an angle = a, a being positive and less than 
c the circumference of the circle, 
then, since g is the m th possible power of a, 
v 
(by Treatise, Art. 59, 60.) length of g — m ih possible power of b 
= (by Art. 18.) ; 
And (by Treatise, Art. 63, 64.) g is inclined to unity at an angle = m . a -H p c ; 
.-. (by Art. 17.) f is the m th general power of a. 
p 
20. Let (ay = where a is any quantity whatever, and m either 
irrational or impossible ; then p = q. 
For let (ay or ( a y ~ ^ 
then, since b = (a\ m , one value of V — m a', 
\pj p 
let b' be that value. 
