Dr. Hirst on Equally Attracting Bodies. 273 



we obtain the equally attracting surfaces 



c'=^!+i^, (C) 



X 



d, = x (P) 



Whence we learn that, the densities at corresponding points being 

 equal, a point is attracted in the same manner bij corresponding 

 portions of an indefinite plane, of its inverse sphere (see II. art. 7), 

 and of any cylinder circumscribing that sphere in such a manner 

 as to have its generators parallel to the plane. To these three 

 equally attracted surfaces may be added, too, the inverse of the 

 cylinder — a surface easily constructed by means of the theorem 

 III. of art. 7. 



31. By the method of art. 23, of which numerous examples 

 have been given, every hypothesis with respect to the forms of 

 the functions '^ and \\ leads, it is true, to a system of four 

 equally attracting surfaces ; but if we propose to find all the sur- 

 faces which attract in tlie same manner as any given one, this 

 method ceases to be applicable. By the theorem of art. 3, how- 

 ever, and by the equation (6) of art. 11, the general integral of 

 the partial differential equation 



(l)'-ii^(|)'=-'^' ■ '«) 



where tan a/t is to be regarded as a given function of 6 and ^, 

 furnishes a complete solution of the more general problem in 

 question. 



35. The integration of (6) presents no difficulties beyond 

 those of ordinary quadrature and elimination when tani/r is a 

 given function of 6 alone. In fact, the condition of intcgra- 

 bility, as well as the equation 



\de} ^m-v^6\d<l^) " ' • ' ^ ^ 



will be satisfied in such cases when 

 du 



, ■ = «= const., 

 d(l> 



du 



-x/'^'-^eK 



(20) 



BO that a solution of (25) is 



Phil. Mag.S. 4. Vol. IG. No. 107. Oct. 1858. T 



