OF ALGEBRAIC QUANTITIES. 
353 
then («y becomes 
771—1 
(I) 
(I) 
becomes 
&c. becomes &c. 
A = 0“ 
B — (i)“ 
cr'-cy 
c = 
m . m — L 
• 0 
&C. = &C. 
77i f , . . m . m — 1 2 . 0 I 
^ = ( J ) • ( 1 + m x + ~ i .2 ' x + &c - j ; 
3c 
Next, let 6 be greater than 
In this case, when x = 0, 
becomes ^ 1 ^ ; 
m . m f i . m.m — 1 2 0 I 
42. Let (f) = g>, where x is a positive quantity, and m a pos- 
sible quantity, and let x and § vary whilst m remains constant ; then § will 
trace out a logarithmic spiral, which cuts its radii vectors at an angle = m. 
For let r be a positive quantity in length = g, 
. m 
then, since = cos m + sin m . — 1, 
/T7 , X X COS 771 + X sin 771 . \/ —1 
f = ( E ) 
x sin m 
= /E\ 
(?) • ( 1 ) 
( E )' 
*, and g is inclined to unity at an angle = .r . sin m, 
r' = x . cos m, 
2 z 
MDCCCXXIX. 
