[ 266 ] 



XXIX. On Equally Attracting Bodies. By Dr. T. A. Hirst. 



[Concluded from p. 177-] 



23. 'T^HE theorem of art. 21 furnishes at once a method of dis- 



J- covering any number of pairs of equally attracting 



surfaces; for the equation (18) of art. 20 may be thus written: 



dv 1 chj 



^=_5|5=M„, . . . ao) 



sin 6 d(p dd 



where /(0, ^) is some function of 6, ^ which we will suppose to 

 be given. The general integrals of these two partial differential 

 equations of the first order and degree will represent two systems 

 of surfaces, such that the normal vector-planes of any surface of 

 the one system will be perpendicular to the corresponding normal 

 vector-planes of every surface of the second system. We may 

 therefore select a surface from each of the two systems for our 

 surfaces {p) and (p,), and thus arrive by the theorem of art. 21 

 at a pair of equally attracting surfaces (?■) and {r^. 



Let us enter a little further into the details of this method. 

 It is well known that the integration of the partial differential 

 equations (19) depends upon that of the ordinary differential 

 equations 



sme^f{e,<f>)- ' I ^2Q^ 



,-^-m0)#=oj 



in such a manner, that if the integrals of the latter be repre- 

 sented, respectively, by 



Y{6, 0) = const., 



Fj(^, ^) = const., 

 the complete integrals of (19) may be written, respectively, thus, 



i; = 2log[^F(F)], 



.. = 21og[^i(F0], 



where "^ and "^j are symbols of arbitrary functions. Thence by 

 art. 21 the general equations of a pair of equally attracting sur- 

 faces will be 



2M = ?; + t;i = 21og[^.%], 



2Mi=w-Vi = 21og[^:%]} 



or, including at once the inverse surfaces »•', »^, to r, »•,, the 



