I 



Dr. Hirst on Equally Attracting Bodies. 311 



diate curve be a logarithmic spiral having the equation 



1 1 a 



— z=v=^ e . 



p 2a 



One of the system of cui-ves cutting this orthogonally will be 

 found, by integrating the equation 



v'l _ V _ 1 



Wj v' m' 



to be 



1 1 -le 



where «i is any particular value whatever of the arbitrary con- 

 stant of integration. 



The pair of equally attracting curves will consequently be 

 represented by the equations 



1 1 „^„ 1 _i.e 



r 2a ^2«j 



rj 2a 2a, 



In the particular case of « = Aj and m = 1, the above reduce them- 

 selves respectively to 



p =2ae~^, 



Pi=2ae^, 



e^ + e-' 



=rc0!i6 = a, 





2 



These curves, for positive values of 9 only, are represented in 

 Plate II. fig. 1 byO??ifl, Oam^, AMO, and A,M,0 respectively. 

 Of the two equally attracting curves AMO and Aj Mj 0, the 

 latter has an asymptote AjB parallel to the prime radius X, and 

 at the distance AB = OA = Afl = «; and the former cuts the prime 

 radius perpendicularly in A, A B and Aj B being corresponding 

 tangents. As is easily seen, both these curves, together with 

 their intermediate curve m a, approach each other and the pole 

 very rapidly. The logarithmic spiral Q B, whose equation is 



r=a \^2^~\ 



is the locus of the foot of the perjjendicular Q from the pole 

 upon the tangent Q in of the intermediate curve Oma, and 

 therefore also the locus of the intersection of the corresponding 

 tangents Q M and Q M, of the equally attracting curves AMO 



