Dr. Hirst on Equally Attracting Bodies. 175 



the problem. The case where Iq or t^ vanishes, here excluded, 

 requires a diflferent, but equally simple mode of treatment. 



19. As an example, let us examine a class of surfaces with 

 which we shall afterwards be occupied, — the class for which -y^ is 

 a function of 6 alone. This class contains as many groups of 

 surfaces as the function i/r of ^ is susceptible of different fonns, 

 and each group includes as many sub-groups of equally attract- 

 ing surfaces as there are different varieties of ■^^ under the same 

 general form. The class clearly includes, too, all surfaces of 

 revolution around the ,r-axis; in fact the general characteristic 

 property of all surfaces of this class is that the attraction of a 

 poi'tion of any one upon the pole is the same as that of the cor- 

 responding portion of the same surface after it has been caused 

 to rotate through any angle around the a^-axis. 



With such properties, however, we shall afterwards be more 

 concerned ; our immediate object is to determine whether, in the 

 class in question, there are one or more groups of surfaces which 

 contain pairs of the kind considered in the preceding articles, 

 and if so, to determine such groups and such pairs of surfaces. 

 By hypothesis, t and t are functions of a> alone, so that the con- 

 dition which, by art. 17, t must fulfil, reduces itself to 



dco^ ' 

 whence, as in art. 16, we conclude that t must have the form 

 T=ma>+ const., 



afid 6 



t = e'^ = 7nAe""^ = mA. tan™ ^ ; 



that is (see art. 17), in the class in question the only groups of 

 the required kind correspond to a value of -^^ of the form 



tan"*^ 

 tan-x/rrrmA .— ^ — -g-, (16) 



sm 



where m and A are arbitrary constants. Further, 



Iq = mA and J^',6?| = /wA^e'"^ ; 

 hence F(^) = Ae'»f+-, and F,(^)=Ae'»f--: 



80 that by art. 15, 



M= — m' = A . tan*"^ . cos (m0-|-«), 

 a 



M,= — M', = A.tan'";r . sin (»i^ -)-«), 



from which we deduce without difficulty the four pairs of sur- 

 faces of the required kind which each group, determined by the 



