276 Dr. Hirst on EquaUy Attracting Bodies. 



whence we deduce 



1— sin'^^.sin^^* 

 This value, substituted in the first of equations (30), gives 



— M = M' = A,sin-'( A^l— sin^^.sin^0). 



The nature of the surface here represented will be at once evi- 

 dent on introducing the angle ^i between the radius vector and 

 the ^r-axis; for since sin ^. sin (^=; cos ^i, the above equation 

 becomes simply 



u' = \0^, or r' = c'e^^j, 



and represents a surface of revolution around the ^•-axis generated 

 by a logarithmic spiral whose pole is at the origin. The surface 

 inverse to r' is of a similar nature, as already remarked in 

 art. 8. 



38. We omit several other cases, where « may be eliminated 

 without difficulty from the equations (30), in order to notice with 

 equal brevity tlie group of surfaces alluded to in art. 19, as being 

 the only one, of the class under consideration, wliich contains 

 pairs of equally attracting surfaces whose corresponding normal 

 vector-planes are perpendicular to each other. Making a = mA 

 in equation (16) of art. 19, we have now 



tan2'» - 



©2 = «2__^. 



so that, by (37), 



The integration here indicated presents no difficulty, and wlien 

 eff'ccted, gives 



F(^,«) = -\/«2 tan2'« f - «2._ ^ tan-' 

 whence, by differentiation. 



a^'tan-'"-— a- 



^-__itan- 

 da. m 



V . } 



Accordingly, by the forraul?e (29), the group of equally attract- 

 ing surfaces under consideration is represented by the equations 



