278 Dr. Hirst on Equally Attracting Bodies. 



the value loa- - ; so that, for these surfaces (?•), the equations (32) 



may be written in the form 



b_ _ ^/siu" g - «^ cos^ ^ + 1 + cos^^ -^ 



r cos 6^ >• • (^^") 



— ■ ■ o ^ =sin2(g)— p), J 



where b and /3 are constants dependent upon a, itself conceived 

 to have a constant value. By means of the latter, the former of 

 these equations may be replaced by 



-cos ^- 1 - cos- e= ± ^sin^ 6 oos 2(<^-/3), 



where for brevity we have made «!= 'v/«^ + 4, so that «i > 2. 

 Passing to rectangular coordinates by means of the formulae 

 r cos 6=a-, r sin 6 cos (0— yS) = 7/, r sin ^ sin {^—/3) =z, it will be 

 found that the characteristic under consideration is the intersec- 

 tion of one of the two surfaces, 



with the surface 



2ux^ + af-2a^7js + az^ = (33) 



Here (H) and (Hj) are hyperboloids of one sheet, each of which 

 has its principal axes parallel to those of the coordinates, a 

 vertex at the origin, and its centre in the a?-axis. The plane 

 [yz), in fact, touches (H) along two generators whose equations 

 are 



z- ~ Vuy+2 

 and (Hj) along the two generators 



y^ + «i + ^. 

 z ~ a ' 



The surface (33) is clearly a cone of the second order with its 

 vertex at the origin, and symmetrically placed with respect to the 

 plane {yz) ; the latter, in fact, cuts the cone in the two gene- 

 rators 



y^'t^, (L) 



z a. 



