MR. WARREN ON GEOMETRICAL REPRESENTATION 
.354 
’ ' COS Jtl ’ * 3 
0 
sin iyi 
§ is inclined to unity at an angle = • r' = tan m . r', 
o o 
which is the property of a logarithmic spiral which cuts its radii vectors at an 
angle = m, 
the curve traced out by § is a logarithmic spiral which cuts its radii vec- 
tors at an angle = m. 
43. Cor. 1.) The logarithmic spiral in the last article will cut the positive 
direction at a distance = 1. 
For let x = 0, then § = /E\° = 1, 
/. one of the values of § is 1, 
.'. the spiral cuts the positive direction at a distance = 1. 
44. Cor. 2.) When m is such that tan m — 0, the spiral becomes a straight 
line ; and when m is such that tan m is infinite, the spiral becomes a circle. 
45. Let a be any quantity whatever, and x any possible quantity, and let 
and let § and x vary while a and p remain constant; then § will trace 
out a logarithmic spiral. 
For let a! = n ^ ~ 1 } where n is positive and m possible, 
then 
m/Ef" ^ “ 1 / 
"(?) U = (“) =f> 
.*. (By Art. 42.) since m is constant, and n x and § variable, § will trace out 
a logarithmic spiral. 
4G. Cor. 1.) The spiral will cut the positive direction at a distance = 1, 
and will cut its radii at an angle = m. 
47. Cor. 2.) § becomes equal to a in its p + 1 th revolution in the spiral, 
reckoning from the time at which it was equal to 1. 
For let b be a positive quantity in length = 0 , and let a be inclined to unity 
at an angle = a, where a is positive and less than c, 
then I > -J- u p c . nj — 1 — 0* 
0 i> 
(O')'-”© 
