Dr. Hirst on Equally Attracting Bodies. 281 



and (— ?•' ) are also two pairs of inverse surfaces, or, as we may 

 express it, (;•) and (?-j), (?•') and {r'-^ constitute two pairs of sur- 

 faces negatively inverse to each other, so that the surfaces of each 

 pair attract equally, but in opposite directions. If, then, by 

 squaring we cause the double signs in (38) to disappear, we shall 

 obtain the equation of a surface composed of (?•) and (rj — or of 

 (r') and (r'j) — which by its attraction will keep the pole in 

 equilibrium, and one-half of which will attract in the same man- 

 ner as the given plane, whilst the other will attract equally but 

 in an opposite direction. In this case, too, the two compound 

 inverse surfaces (r, r^ and (r', r'j) will be of the same kind, and 

 differ only in position ; in other words, both may be represented 

 by the single equation 



{ai\l + cos- 6) + ?•' siu^ 6 cos 2(jf)-c' cos e'\'^ = r'^-\], 

 which by simplifying, dividing throughout by — 4cos^, setting 

 c in place of ^, and omitting the accent of r, becomes 



/;V cos e + c?-[fl(l + cos2 6) + siu2 6 cos 2<^] -c^ cos ^=0. . (39) 



In rectangular coordinates the surface here represented has the 

 equation 



[^2(a;2 + 2/3 ^ z-~) _ c^] ^^. + c \2ax'- + (1 + a)y''-{\ - a)z"-'\ = 0; (40) 

 or, as it may be otherwise written, 



^V ^v 



{c{\-a)-b'^x]x {c{\ + a)+b\x'\x~ ' ^^ 



44. This surface of the third degree, which we have thus 

 found to attract the origin in the same manner as any plane 

 perpendicular to the a?-axis, is altogether contained between the 

 two planes 



x=^^{l-a), 



the former of which cuts the ir-axis, say on the right of the ori- 

 gin O in a point C, and touches the surface along a line CZq par- 

 allel to ther-axis; whilst the latter cuts the ^r-axis in a point C, 

 on the left of 0, and touches the surface along a line CjY, par- 

 allel to the ?/-axis. Eveiy plane parallel to, and intermediate 

 between the foregoing outs tlie surface in a hyperbola whose 

 centre is in the A'-axis, and whose vertices are in the plane {xy) 

 or in the plane {xz), according as the plane of the section is on 

 the right or left of the origin. In the ])lane [yz) itself, this 

 hyperbola degenerates into two lines OA, OB, equally inclined to 



