Dr. Hirst on Equally Attracting Bodies. 173 



and hence M + iMi = F(o) + J^) 1 Q4^ 



M — mj = ri(6)— i0).J 

 In our case, therefore, where u and u^ necessarily represent real 

 functions, F and F^ instead of being perfectly arbitrary are so 

 related that one becomes identical with the other on changing i 

 into —i; in other words, in order that the surfaces represented 

 by the functions u and m, of w and </> may be equally attracting, 

 and at the same time have their corresponding normal vector- 

 planes perpendicular to each other, it is necessary and sufficient 

 that one should be the real part of, and the other the coefficient 

 of i in some function F(ft) + i</>). 



16. The simplest example of such a pair of surfaces is 



M=mft) = log (tan"'^jj 



^"^ - r = c.tan'»H, 



The former of these is the only surface of revolution which is 

 compatible with the conditions under consideration ; for in the 

 case of u being a function of d alone, the equation (13) reduces 

 itself to 



dco'^ ' 

 whose integral is, clearly, 



u = mco + A, 

 where m and A are arbitrary constants. The surface (?•,) would 

 be generated by a circle of variable magnitude, moving so that its 

 plane always passes through the ,r-axis, its centre always coin- 

 cides with the origin, and its radius with a radius vector of a 

 logarithmic spiral traced in the plane {yz) with the origin for 



pole. 



17. We may always decide at once whether, amongst the 

 group of surfaces which attract in the same manner as any given 

 one, there exist conjugate pairs whose corresponding normal 

 vector-planes are perpendicular to each other, and in the case of 

 their existence we can easily detect them. In fact, by the fun- 

 damental theorem of art. 3, the acute angle ^]r between the nor- 

 mal and radius vector, regarded as a given function of 6 (or to) 

 and (f>, determines the group in question, inasmuch as it is iden- 

 tically the same for every surface which this group includes. 



Let us suppose / 



tan yjr = -. — ^, 

 ^ sin a 



