OF ALGEBRAIC QUANTITIES. 
357 
now, if we substitute for p successively 0, 1,2, &c., also — 1,-2, &c., the 
values of ^E^ mw+pcm ^ ~ 1 will be in geometric progression, 
the values of f are in geometric progression. 
54. Cor. 1.) Hence all the values of § are radii vectors of the same loga- 
rithmic spiral. 
55. Cor. 2.) If m be impossible or irrational, § will have an infinite 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. 
56. Any geometric series being given, it is required to find quantities a and 
m , such that, a 1 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 = 
then b r — 
Now 1' = c f^/ — 1 , where c is the circumference of the circle, 
r' = m c ^/ — 1, 
.-. m — — — .*. m is known ; 
c V — l 
Now b = , 
• • Cl C 1 . yl , 
P 
2 
57. Cor.) If the series be of the form 1 , r, r , &c., we may take b = 1 and 
b' — 0 ; then we shall have a = 1, and a 
shall have a = 1, and a 1 — l c ^ \ 
