828 .IN l/.r/V GEOMETRY [Oft 



196. The parabolic spiral. This curve is tra--d l>y a 

 point \\hirh moves around a fixed point in ;i plane in surh 

 a way that the squares of any two radii vectores are in tin- 

 same ratio as are the angles \\hirh they form with the 

 initial line. 



From this definition it follows that, tin- equation of the 



curveis 



where A; is a constant. 



This equation shows that the curve begins at the center 

 when = 0, winds round and round 

 this point, always receding from it, 

 the radius vector becoming infinite 

 when 6 becomes infinite, i.e.. \\ h< n 

 -R it has described an infinite number 

 of spires. 



The locus of equation (1), for 

 positive values of p 1 is represented 

 Fl0 - 137 in Fig. 137.* 



197. The lituus f or trumpet. This curve is traced by a 

 point which moves around a fixed point in a plane in such 

 a way that the squares of any two radii vectores are in the 

 same ratio as the reciprocals of the angles which they form 

 with the initial line. 



From this definition it follows that the equation of the 



k 



curve is & = , (1 ) 







where & is a constant. 



This equation shows that the curve begins at infinity, 

 when = 0, and winds round and round the center, always 



See also Rice and Johnson's Differential Calculus, p. 307. 



t This curve was invented and named by Cotes, who died in 1710. 



