Dr. Hirst on Equally Attracting Bodies. 307 



brace both, we may at oace conclude from the above theorem 

 that all right lines equidistant from the pole, together with the 

 circle which they envelope, constitute a system of equally attract- 

 ing curves. This conclusion is merely a more general expression 

 of a theorem long known, and communicated some years ago to 

 the Cambridge and Dublin Mathematical Journal by Professor 

 Joachimsthal. 



The condition that the corresponding elements of the two 

 curves ,.=/(^), ^^=/^(^) 



may attract the pole equally, is also expressed by the differential 

 equation 



u^ + u'^=u\ + u'\, (1) 



which must be fulfilled for all values of under consideration. 

 This equation may be written thus, 



or, if we make 



thus 



u' + u'i u'—u\ 



M + Mj U — Uy 



= -\; .... (2) 



2v=u^-u{\^ ^3^ 



2w, 



HJL = -1, ....... (4) 



the last of which equations will be fulfilled when 



- =W), 



1 



where F(^) is an arbitrary function of 6. Integrating these 

 equations, and introducing two arbitrary constants c and Cj, we 

 find 



CF(8)rfe "1 

 »=c«J L (5) 



-f— I 



By addition and subtraction in accordance with equations (3), 

 wc thus deduce the following general equations of a pair of 

 curves whose con'esponding elements attract the pole equally : 



L (6) 



' »■, -J 



Before considering a few special cases of this general formula, 

 Y2 



