Dr. Hirst on Equally Attracting Bodies. 269 



ing conoidal surfaces corresponds to the hypothesis 



^ (y\ ^ ax + bij 

 ^\x/ a^x + b^y' 



where a, b, a^, i, are arbitrary constants^ for then the equations 

 (24) become 



^^^ax + by_ 



a^x + b^y' ^ ' 



a^ + b^ (HO 



' ax + by ' ^ ^' 



and represent two equilateral hyperbolic paraboloids having a 

 common generator — the 5:-axis — passing through their vertices 

 and the attracted point. 



Any plane through the s-axis, e. g. 



ax + by _ a 



a^x-\-b{y /3' 

 where a. and /? are constants, touches the paraboloids (H) and 

 (Hj) at points onthe^r-axis whose distances from the origin are 



s = c^, and Si = c^-, 



respectively, so that the product 



^ . 2"! = c . Cj = const. ; 

 that is to say, these points of contact form a system of points in 

 involution, the origin or attracted point being the centre of the 

 system. The two surfaces, therefore, are so situated with respect 

 to each other, that the asymptotic plane, through the common 

 director, of either surface touches the other in the attracted 

 point; and they have, moreover, two generators in common 

 which pass through the double points of the system in involu- 

 tion, and are situated at the distance V c .c^ on each side of the 

 attracted point. These coincident generators are, of course, real 

 or imaginary according as c and c, have like or unlike signs. 



The result of the foregoing investigation may be thus ex- 

 pressed: — IVJien any two equilateral hyperbolic paraboloids have 

 in common a director through their vertices, and at the same time 

 are so situated that the points of contact of planes 2Jassing through 

 this director constitute a system of jioints in involution, the centre 

 of this system will be attracted eqiially by corresponding portions 

 of both surfaces, provided the densities at corresponding points are 

 the same. 



29. In order that the accuracy of this theorem may be per- 

 fectly evident, we ought to remark that (H) or (Hj) is the most 

 general form of the equation of an equilateral hyperbolic para- 



