174 Dr. Hirst on Equally Attracting Bodies. 



where Ms a known function of tw and <p. The equation (6), 

 art. 11, will then resolve itself into 



and assuming the existence, in the group determined by t, of 

 pairs of surfaces of the required kind, the function u which re- 

 presents one of such a pair will have the form 

 2M = F(a) + «^) + F,((a — 2^). 

 Differentiating, we find 



2 ^ = i¥{<o + i^) - iF'i (o) - i<f>) ; 



hence, squaring and adding, 



/2 = F'(a)+z<^).F',(a)-«(^), .... (15) 

 or 



2t= log t^ — log F'(ft) + i^) + log F'l (o) - 10) ; 



that is to say, if pairs of surfaces of the required kind exist, 

 T=log/ must have the same form as u, and consequently fulfil 

 the condition ^2^ ^2^ 



18, When the given function t satisfies this condition, the 

 pairs of equally attracting surfaces in the group i^^), which have 

 their corresponding normal vector-planes perpendicular to each 

 othei', may usually be determined with great facility. To do 

 so it is only necessary to determine F' and F'j — and thence F 

 and F, — from the identical equation (15). If in the latter we 

 set &) = <^ = 0, and denote the result of this substitution in / by 

 /(), we have 



^%=F'(0).F\(0), 



which relation, since F' and F', differ from each other only in 

 the sign of i, will be fulfilled in the most general manner by 

 setting 



F'(0)=/oe'«, F'i(0)=^oe--, 



where a is a real arbitrary constant. Again, setting t<) = ?0=^, 



and denoting by ^j the result of this substitution in t, 



^S = F'(^).F\(0) = V«.F'(?), 

 and 



F(?) = ^hVf + const. 

 By changing i into —i we at once obtain Fi(f), and thus solve 



I 



