OF ALGEBRAIC QUANTITIES. 
341 
4. Cor. 1.) f d -f = i 
For (by Treatise, Art. 168.) — v + 0 — 1. 
5. Cor. 2.) If v + 0 \f — 1 = §' ; then v + 0-{-pc*/ — lis also a value 
of where p is any whole number, either positive or negative, and c is the 
circumference of a circle whose radius = 1. 
For, since § is inclined to unity at an angle = 6, it is also inclined to unity 
at an angle = 6 + p c, 
(by Art. 3.) v + 0 + p c . — 1 = §' 
6. Cor. 3.) Hence, if 0 be positive and less than c, £ = v + 0 + p c , — 1. 
p 
7. Cor. 4.) Hence ^ = §' + p — q . c — 1. 
p ? 
For g' = v + 0 + pc. — 1, 
p 
and §' = v + 0 + q c . — 1, 
q 
§' = §' + p - q • c V -T. 
p q 
8. Cor. 5.) Hence §' = g' + p c — 1. 
p 0 
9. Cor. 6.) If § be a positive quantity, §' = possible hyperbolic logarithm of §. 
0 
10. Cor. 7-) Hence, if § be any quantity whatever, and r be a positive 
quantity equal in length to §, and § be inclined to unity at an angle — 0, 0 being 
positive and less than c\ §' = r' + 0 p c . y/ — 1. 
p 0 
1 1 . Def.) Let a and § be any two quantities whatever, then -|j is called the 
general logarithm of g in a system whose base is a. 
12. Cor. 1.) If a be the base of a system of general logarithms, then the 
v 
general logarithm of a in that system is 1 . 
p . 
13. Cor. 2.) Let E be a quantity such that E' = 1, and let § be any quan- 
0 
tity whatever ; then §’ = general logarithm of § in a system whose base is E. 
