Dr, Hirst on Equally Attracting Bodies. 371 



have the conditions 



(« + a]7) cos \— (yS + y3,7) sin X _ a, 

 (a + a,7) sinX+ ()3 + ;8j7) cosX, i/ 

 ajCOsX — ySjsin\ a 

 «! sinX+/3j cosA, b' 



from which we deduce 



u^b-P^a (a + «i7)Si — (/S + /3,7)«, 



tan\ = 



the magnitude of X, therefore, is determined. That of 7, after a 

 few easy reductions, is found to be 



_ _ (a^-gi6)(«aj + /3/3i) + Hi + M,)(«/3i-aj/3) 

 "^ {ab,-a,b){u\ + ^\) 



31. It would not be difficult to establish the theorem of art. 

 28 by proving, directly, that the surfaces (H) and (Hj) cut each 

 ray of a pencil, whose centre is the attracted point, at equal 

 angles; we omit this demonstration, however, as well as the 

 establishment of a similar theorem for all the conoidal surfaces 

 of art. 27, in order to notice briefly the two surfaces inverse to 

 the paraboloids (H) and (Hj), and which attract in the same 

 manner. Their equations might, of course, be just as readily 

 found as were those of (H) and (Hj) ; we prefer here, however, 

 to confine ourselves to the generation of the surfaces in question, 

 as suggested by the theory of inverse surfaces. 



If in the axis of a pencil of planes we take a system of points 

 homographic with that pencil, we know that an equilateral hy- 

 perbolic paraboloid is generated by a right line moving so as 

 always to cut the axis perpendicularly in the point which corre- 

 sponds to the plane of the pencil in which such line is situated. 

 The inverse of this generating line, with respect to a fixed 

 point of inversion in the axis of the pencil, is a circle pass- 

 ing throiigh 0, and having its centre in the axis ; further, the 

 centres of all such circles clearly form a system of points ho- 

 mographic with the first system and, therefore, with the pencil 

 of planes. So that having again taken a suitable system of 

 points in the axis of, and homographic with the pencil of planes, 

 the inverse of the above equilateral hyperbolic paraboloid would 

 be generated by a circle of variable magnitude moving so as 

 always to pass through the fixed point 0, and to have its centre 

 at that point of the axis which corresponds to the plane in which 

 the circle is situated. The surface, generated in the same man- 

 ner, which attracts equally is so placed that its generating 

 circles all pass through O, and have their centres in the same 



