320 Dr. Hirst on Equally Attracting Bodies. 



"With respect to the gi'oup of convex curves, they intersect all 

 the spirals between infinity and the limiting curve at positive 



angles of inclination, which increase continuously from o- — ■Cq 



to TT— CTq. They all cross the given spiral (19) at the angle 

 TT— 2'B7o, and terminate in the limiting spiral in the same man- 

 ner as the concave curves. 



These properties, as well as some others, are easily deduced 

 from the equations (22). If in the latter we put r=p, we find 

 that the concave and convex curves, corresponding to the same 

 value of c, terminate at the same point of the limiting curve, viz. 

 that for which 



To find the intersection points with the given spiral (19), we 

 have merely to make r=m^p, when (32) becomes 



m^^o=mlog c( — + — ) + cos-* — , 

 so that for all concave curves, 



and for any convex curve, 



?w^,^o= '"log [mc) — m log — -' — cos~' — ; 

 that is, 



«,-«o=i(»losf+cos-.i). 



The quantity in parenthesis is easily seen to be always positive 

 for positive and real values of m, hence ^, < ^^j and as the dif- 

 ference between these angles is independent of the arbitrary con- 

 stant, we conclude that the given spiral (19) and its limiting 

 curve intercept, upon all the equally attracting convex curves, 

 arcs which subtend equal angles at the pole. '\Vhen?- = oo, 

 equation (22) gives 



hence 



^>-^"=^,('"^'^'"+?)- 



The least possible value of m log m, m being always positive, is 

 - log - = ; consequently for all convex curves 



