Dr. Hirst on Equally Attracting Bodies. 321 



aad every such complete curve subtends a constant angle at the 

 pole. With respect to any concave curve, its initial radius vector 

 corresponding to 9^, meets the curve again or not, according as 



^m<gr. ^vhen7w'"=e2 (or »i = 2'1073 nearly). This radius 



vector is parallel to the asymptote. The asymptotes of any two 

 concave and convex curves which meet in the same point of the 

 limiting curve are at the distances 



„ m^ m TT^ Q m^ _ m ^ 

 C«^ . em^ * 2 , Cn^i . e m^i * 2 



respectively, from the pole, but on opposite sides of the same with 

 respect to the radii vectores parallel to these asymptotes. The 

 inclination of the latter towards each other is the same for all 

 pairs of such curves, and equal to 



TO J 



In fig. 4, which corresponds to the particular case of m=l, 

 »i-j = 2, O M M, is the given spiral, cutting all its radii vectores 



TT 



at an angle equal to — ; O/Lt/ij is its limiting curve, which, as 



we know, is a spiral of the same kind. Any point ^ on this 

 limiting curve being taken, for which /LtOX = ^j, the two curves 

 which meet in the same are easily traced by means of the equa- 

 tions (21) and (21«). Of these the concave curve consists of two 

 branches, m-^imii^ and fj,m"cf), which meet each other and the 

 given spiral only in the pole. The convex curve /xNm'^ crosses 

 the given spiral perpendicularly in N ; the asymptotes tn^P^ and 

 m'gP, of these curves are at right angles to auothei', in accord- 



TT .... 



ance with the general formula -g- for their mclination. We may 



notice, lastly, that when m is allowed to diminish indefinitely, 

 the given spiral (19) coincides ultimately with its limiting cm've 

 and the circle whose centre is the pole and radius a. At the 

 same time m, approaches unity as its limit, and the equations 

 (22) of the two groups of equally attracting curves reduce them- 

 selves to 



6 — a,= + C0S"'-, 

 ?' 



where a is the arbitrary constant. As these equations repre- 

 sent right lines touching the above circle, we are led once more 

 to the case already frequently mentioned. 



When the given attracting curve is one of double curvature, 

 we may conceive it to be described upon the surface of a cone 

 whose vertex is the pole, and whose direction is the curve itself. 



F/iil. May. S. 4. Vol. 13. No. 87. May 1857. Z 



