273 Dr. Hirst on Equally Attracting Bodies. 



line; in fact, the centres of generating circles, in the same plane, 

 form a system in involution whose centre is 0. 



It is scarcely necessary to remark that there are other ways 

 of generating the inverse of an equilateral hyperbolic paraboloid ; 

 for example, since the latter surface may be generated by a line 

 moving along two directors so as constantly to remain perpen- 

 dicular to one of them, the inverse surface will be generated by 

 a circle moving so as always to pass through a fixed point, to 

 have its centre in a fixed Hue through that point, and to have 

 as director any other circle whatever passing through the same 

 fixed point. 



32. As a last example of equally attracting conoidal surfaces 

 represented by the equations (24), art. 27, we may mention the 

 case where one is a so-called skew helicoid. As is well known, 

 the curvilinear director of this surface is a helix on the surface 

 of a common cylinder, the rectilineal director, to which the ge- 

 nerator is always perpendicular, being the axis of the cylinder — 

 in our case the ^■-axis. The equation of such a surface is easily 

 found to be 



z=kai&xi-'{y^, (S) 



where a is the radius of the circular base of the cylinder, and k 

 the tangent of the angle at which the helix cuts the plane of that 

 base. By art. 27, therefore, the equation of the equally attract- 

 ing conoidal surface is 



(S,) 



-(f)' 



tan 



The director here is an equilateral hyperbola traced upon the 

 surface of the cylinder ; one of its asymptotes is the generator 

 of the cylinder through the point where the helix cuts the base, 

 the other is the base itself, as is at once evident from the fact 

 that the ordinates of the surfaces (S) and (S,) have reciprocal 

 values. 



33. The particular cases of the general equations of art. 23 

 are, of course, inexhaustible ; Vv'c will limit ourselves to one more 

 which, from its simplicity, possesses some interest. By setting 



and selecting the formulae for ?■' and r\, given in (23) of art. 26, 



