Dr. Hirst on Equally Attracting Bodies. 379 



one of which, it will be observed, coiDcides with a generator of 



(H), whilst the other coincides with a generator of (Hi). 



It can be easily shown, however, that if, by squaring, we cause 

 the radical sign to disappear, the two equations (32 ) will then 

 represent a surface composed of the inverse surfaces (r) and (r ). 

 The characteristic of this more general surface will be the inter- 

 section of the cone (33) with both hyperboloids (H) and (Hj) ; 

 in other words, this characteristic will consist of two right lines 

 (L) in the plane {ijz), and of two curves of double curvature of 

 the third order, all of which pass through the origin*. 



41. With respect to particular cases of the surfaces represented 

 by (32), we remark, in the first place, that the hypothesis a = 

 leads to the plane and its inverse sphere, 



r _d ^ 1 

 c 7^ cos 6' 



as may be best seen by treating, directly, the original differential 

 equation (25). 



42. As a second hypothesis, let 



where a and b are any constants less than unity, and fulfilling 

 the relation 



a^ + b''~ = l. 



In accordance with this hypothesis, we easily find for /(«) the 

 value 



/(«) = ^ sin-> ( ^^) _ log [«( ^P^^N^-2«)] . 



By substituting these values of /(«) and f'{u), reducing, and 

 neglecting constants which do not affect the generality of the 

 solution, the equations (32) become 





cos^(a/6V+4-2«) 

 1 + cos^ e 



\ . (34! 



= a. r-2 ^~ -f fla cos 2^ + 'v/6^«^ + 4 . sin 2^, | 



from which « may be eliminated without difficulty. In fact, 



* A valuable memoir, by M. Chasles, on these curves of double curva- 

 ture of the third order will be found in Liouville's Journal (2'"° ser. vol. ii.). 

 We may remark, too, that iu our case both these curves are situated ujion 

 a surface of revolution of the third order, the generating curve of which 

 has the equation 



(ar'+j/=)(oe-i a - 2c)=C«(2a;— c). 



