320 - M I ION. 



Thus < (a) is constant for all values of o, that is 



r a = a constant = k say. 

 r au 



'Phis result ini^'ht have been anticipated : if , xpn-sscs that 

 dements of tfafi curve which subtend equal infinitesimal angles 

 at tlic pole exert equal actions on the particle there. 



Therefore i* + ** = &V ; 



this leads to either -. = 0, or else 



The former supposition makes r constant, and so gives a 

 circle. Taking the latter, and putting - for r we have 



so that (n - 1) 6 + C= sin" 1 -j- , 



where (7 is a constant. 



Therefore n _, = k sin {(n - 1) 6 + C\. 



If 71 = 2 we obtain 



l=rsin(0+ <7), 

 which is the equation to a straight line: see Art. 204. 



n = 3 we obtain 



1 =&r* sin (20 +C), 



which i tii- equation to a rectangular hyperbola, the pole 

 being at the centre. 



