282 Dr. Hirst on Equally Attracting Bodies. 



the coordinate axeSj and having the equations 



^=+ ^^~<^ , 

 ^ ~ V\Ta' 

 When r = 0, the equation (40) resolves itself into 



[c{l + a) + hH'\\h\:i" + if)-c[l-u)x'\=Q; 

 and when y=-0, into 



[c(l -a) -b^s] [62(.r2 + ^2) _,. ^(i + g)^^,] ^q ; 



that is to say, tlie plane {xy) cuts the surface in the line C,Yo, 

 and in a circle (C), described upon OC as diameter, whilst the 

 plane {xz) cuts the surface in the line CZq, and in a circle (Cj), 

 described upon OCj as diameter. 



Again, planes through the <r-axis and the lines OA, OB cut 

 the surface in two curves (0), (Oj) whose common polar equation 



is found^ from (39), by setting tan^^=^j , i. e. cos2(^ = «, 



to be ^,2/.2 cog + 2acr -c'^ cos 6 = 0; 



which equation, expressed in rectangular coordinates x, 77 in the 

 plane of either curve, becomes 



{x^ + V^){b^x + 2ac) = c''x. 



Each section (0), (Oi), therefore, is a curve of the third degree 

 passing through the circular points at infinity and having two 

 coincident foci in the origin 0. Each curve, too, consists of an 

 oval and a serpentine branch, each of which is symmetrically 

 placed with respect to the .2'-axis, and the negative inverse of the 

 other with respect to a circle, around the origin as centre, with 



the radius j. The oval, moreover, is on the right of the plane 



{ys), lies altogether within this circle f y j, and passes through 



the points 0, C ; whilst the serpentine branch is altogether on 



the left of the plane {yz), and without the circle ij); it passes 



through the point Cj, and has for asymptote the line 

 b'^x + 'iac = 0. 



The surface under consideration, therefore, may be conceived 

 to be generated by a hyperbola, of variable form, moving along 

 the circular cubics (0), (Oj), as dii'ectors, in such a manner that 

 its ])lane remains constantly parallel to the plane {yz), and its 

 vertices constantly upon one of the circles (C), (Cj). 



45. The equation (41) represents as many surfaces, attracting 

 in the same manner as the given plane, as there are different 

 valuea of a and b compatible with the condition a^ + b^=l (art. 



J 



