OF ALGEBRAIC QUANTITIES. 357 



now, if we substitute for p successively 0, 1,2, &c., also —1,-2, &c., the 

 values of /e\">«+j'<:»» V- i ^j^ ^g jn geometric progression, 



.-. the values off are in geometric progression. 



54. Cor. 1.) Hence all the values of f are radii vectors of the same loga- 

 rithmic spiral. 



55. Cor. 2.) If m be impossible or irrational, f will have an inj5nite number 

 of different values ; but if m be rational the values will recur, and the number 

 of different values will be equal to the denominator of m, when m is expressed 

 as a fraction in its lowest terms. 



66. Any geometric series being given, it is required to find quantities a and 

 m, such that, a"* may have values equal to each of the terms of the series. 

 Let b be any term of the series, and r the common ratio, 



and let b = (a\"^, 



... . = (;)"^ 



Now 1' = c ^ — 1, where c is the circumference of the circle, 



•■ ((!)7 



= m c ^ — I, 



r 



' = m c >y — I, 



m = — . — - , .-. m is known ; 



C V — \ 



Now b = (a 





/. V z=. m d =■ — , . d, 



I ^ 



.-. a = c ^ — 1 . y, 



p 



J v_ 



:. a = /^EV ~ ■'',.•. a is known. 

 57- Cor.) If the series be of the form \,r,r, &c., we may take ft = 1 and 



r' 



b' = ; then we shall have a = 1, and a™ = 1* . 



