Dr. Hirst on Equally Attracting Bodies. 317 



the corresponding points of all the equally attracting curves of 

 both groups will also be points of inflection. If the given curve 

 has an apse, it will be touched in that point by its limiting curve, 

 which will itself have an apse. The corresponding points of the 

 equally attracting curves will in general be cusps, which will 

 become more and more acute as they recede from the pole. In 

 general, too, the curves of the same group never intersect one 

 another; were they in any exceptional case to do so, then the 

 curve enveloped by them would also be one of the series of 

 equally attracting curves, and would correspond to a singular 

 solution of (14). As an example of this, we have the well-known 

 case when j9 = a. The general integral of (14) is then the equa- 

 tion of a right line 



r cos (a + ^) = a, 



where a. is an arbitrary constant. The equation of the circle 

 around the pole with radius a, or 



r—a, 

 is a singular solution. 



As u is necessarily less than &>, we may put 



« = (i>cos'or, (17) 



where ct is evidently the variable acute angle between the radius 

 vector and the perpendicular on the tangent. Differentiating 

 (17) according to 6, we find 



m' = Q)' cos •or — (bct' sin m ; 



so that by substitution the equations (16) of the two groups of 

 equally attracting curves correspond respectively to the upper 

 and lower signs in 



to'cosCT— wiir'sinOT + wsin ■BT = 0. . . . (18) 



One case of some interest, where the vai-iables in this equation 

 are easily separated, is when the given curve is a portion of a 

 logarithmic spiral whose equation is 



?-=«e"'« (19) 



From this we easily deduce 



-=(»= — 6-""^ 6)'= Le-»i9 ^ ^ (20) 



pa a ^ ' 



where for brevity we have made 



m,= V\ + 



mr 



Substituting the values (20) in the equation (18), and dividing 



m 



the latter by ^ e""*®, it becomes 



7M cos •or 4- ■ct' sin TO + sinCT=0, 



