354 MR. WARREN ON GEOMETRICAL REPRESENTATION 



^ cos TO '^' 







sm7n 



• -1 • 1 oil* '/* I r 



.-. f is inclined to unity at an angle = ^^^ . r = tan m .r, 



o 



which is the property of a logarithmic spiral which cuts its radii vectors at an 

 angle = m, 



.-. the curve traced out by f 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 § = /EV = 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 w = 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 



^a/ = f, and let f and x vary while a and p remain constant; then § will trace 



out a logarithmic spiral. 



For let a' = n /E\"' '^ -^, where n is positive and m possible, 



then ((«)•)' =»x(E)-^-. 

 ■•• (?) *-^ = = '' 



.-. (By Art. 42.) since m is constant, and n x and f variable, § will trace out 

 a logarithmic spiral. 



46. 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.) f becomes equal to a in its j9 + 1* revolution in the spiral, 

 reckoning from the time at which it was equal to 1. 



For let i be a positive quantity in length = a, and let a be inclined to unity 

 at an angle = a, where a is positive and less than c, 

 then b'-{-u + pc.^— 1 = a' 



