313 Dr. Hirst on Equally Attracting Bodies. 



and A, M, 0. Every line m■^ through the pole is divided har- 

 monically in the points 0, M, m, Mj, and 



J_-_l_ JL 



OM~Om^O/«i' 



J__J 1_ 



OMi ~ Om Owi' 



It is scarcely necessary to allude to the analogy between the 

 equations of these equally attracting curves, as expressed by 

 means of the so-called hyperbolic sine and cosine, and those of 

 their corresponding tangents AB and AjBi, whose corresponding 

 elements are also equally attracting. 



In precisely the same manner, when the intermediate curve 

 has the equation 



p= Ccos"- 

 r 



n ■ < 



p, = U|Sm" 

 n 



and the equally attracting curves are 



1 1 



(10) 



+ ■ 



Ccos"- C,sm»- 

 n n 



11 1 



Lcos"- C.sm™- 

 n n 



(11) 



where n may have any value whatever. By making 



n=-l, C=-^, Ci = 

 cos a 



the above equations become, respectively, 



pcos^= , p, sin^ = 



' cos a ' oiij I* 



rco%{6-{-u) = a, 7\cos{6 — u) = a, 



all of which are those of right lines. The first of these lines is 

 perpendicular, and the second parallel to the prime radius; the 

 third and fourth, whose corresponding elements attract the pole 

 equally, are equidistant from the same, the angles they make 

 with one another being bisected by the prime radius and the first 

 or intermediate right line. 



If we make ?«= — 2, C — Ci = 2a, C-f C,= — 2c, and assuming 

 fl> c, b'^=a^—c^, the equations (10) and (11), after a few simple 



