ANALYTIC GEUM1.J /. ) 



L ( ... 



way that any two r.i-lii 

 vectores are in the same 

 ratio as an- tin- angles they 

 make with the initial line. * 

 From this (Idinition it 

 follows that the <-<|ii;itn n 

 of the curve is 



P = M, . . . (1) 

 where k is a constant. 



This equation shows t hat 

 the locus passes through the origin, and that the rac'ms 

 vector becomes larger and larger without limit as the num- 

 ber of revolutions increases without limit. Moreover, it 

 (jp r 0j) be any point on the curve, and if (/? 2 , B l + - TT) he 

 the corresponding point on the next spire, then 



whence f> 2 = p l + 2 kir ; 



but 2 krr = OA, hence the distance between the successive 

 points in which any radius vector meets the curve is constant ; 

 it is always equal to the radius of the measuring circle. This 

 follows also directly from the definition. 



The locus of equation (1), for positive values of is rep- 

 resented in Fig. 135 ; for negative values of the locus is 

 symmetrical with the part already drawn, the axis of sym- 

 metry being the line LF. 



195. The reciprocal or hyperbolic spiral. This curve is 

 traced by a point which moves about a fixed point in a 

 plane in such a way that any two radii vectores are in iln- 



This curve may also be defined thus : It is the path traced by a point 

 which moves away from the center with uniform linear velocity, while ito 

 radius vector revolves about the center with uniform angular velocity. 





