OF ALGEBRAIC QUANTITIES. 341 



4. Cor.l.)/f = f' 



For (by Treatise, Art. 168.)^^^ = v + 6 ^ — 1. 



5. Cor. 2.) Uv + 6 ^ — I = g' ; then v + 6 + pc ^ — 1 is also a value 

 of f', where p is any whole number, either positive or negative, and c is the 

 circumference of a circle whose radius = 1 . 



For, since g is inclined to unity at an angle = 6, it is also inclined to unity 

 at an angle — 6 ■\- p c, 



.-. (by Art. 3.) «^+ 6 + p c . J — 1 = f' 



6. Cor. 3.) Hence, if 6 be positive and less than c,^ ^=-v ■\- 6 -\- pc . fj —\. 



p 



7. Cor. 4.) Hence f ' = g.' -|- jo — q . c ^ — \. 



p 1 



For / = w + -\- p c . s/ — 1, ' |%f •- . ,'';i»'- . 



;' ''* ' 



and g' = V + 6 + qc . ^ — 1, 



p q 



8. Cor. 5.) Hence ^ = ^ -\- p c J — \. 



P 



9. Cpr. 6.) Iff beapositive quantity, g[ = possible hyperbolic logarithm of p. 



10. Cor. 7-) Henc€?, if g be any quantity 'whatever, and r be a positive ^^ 

 quantity equal in length to §, and f be inclined to unity at an angle =. 6, 6 being 



positive and less than c; ^ = r^-\-6-\-pc.^— \. 



1 1 . Def.) Let a and g be any two quantities whatever, then ^ is called the ., - 

 general logarithm of f in a system whose base is a. 



12. Cor. 1.) If a be the base of a system of general logarithms, then the 



p 



general logarithm of a in that system is 1 . 



p 



13. Cor, 2.) Let E be a quantity such that E' = 1, and let g be any quan- 







tity whatever ; then g' = general logarithm of ^ in a system whose base is E. 



