OF ALGEBRAIC QUANTITIES, 343 



But § is inclined to unity at an angle = 6 + q c, 



.*. f is inclined to unity at an angle = m . a -\- p c ; 



.'. § is in length = (bY^, and is inclined to unity at an angle = m .cc -\- p c. 



18. Let a be a positive quantity, and let § be the ni'^ possible power of a ; 

 .then g will also be the m* general power of a. 







For since a is a positive quantity, and f the m"" possible power of a, (by 



o 



Treatise, Art. 65.) f is a positive quantity; 



Also, from the nature of possible logarithms, since f is the m*'' possible power 

 of a ; possible hyperbolic logarithm of g> = m X possible hyperbolic logarithm 



o 



of a, 



that is (by Art. 9.) ^ ' = m a', 



O 



.•. (by Art. 15.) §> is the m"" general power of a. 







1 9. Let a be any quantity whatever, and let g be the m"" possible power of 



a, then § will also be the m^ general power of a. ■ ..^ » 



p ^ ■ p 



For let 6 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, 



tlien, since g is the w* possible power of a, 



p 

 (by Treatise, Art. 59, 60.) length of g- = m* possible power of h 



= (by Art. 18.) aV 



And (by Treatise, Art. 63, 64.) g is inclined to unity at an angle = m . a -\- pc; 



.-. (by Art. 17.) g is the tn^^ general power of a. 



p 



20. Let ^aV"^ = (aV^, where a is any quantity whatever, and m either 



irrational or impossible ; then p = q. 

 For let /aY* or (a\^'- = b, 



then, since b = (aV', one value of b' = m a', 

 let b' be that value. 



