'=^\/«' 



Dr. Hirst on Equallij Attracting Bodies. 377 



1 



tau^'"3 -«-+/(«)-«/'(* 



1 /v/anan'"^ 

 0=--tan-i'l !_ ^ 



tan^-^-a^* ' • ^^^^ 



tan-i\ ^ A l+<^.f./(«)J 



from which, as before, a. is to be ehmiuated as soon as a parti- 

 cular form for /has been chosen. 



The hypothesis /(«) =/'(a) =0 leads at once to the first of the 

 pair of surfaces given at the end of art. 19; the second surface 

 of the pair, whose corresponding normal vector-planes are per- 

 pendicular to each other, results from the hypothesis /(«) = « .—, 



i« 



and consequently, by art. 36, differs only in position from the 

 first surface. 



39. We will treat at somewhat greater length the more inter- 

 esting case of the group of surfaces which attract in the same 

 manner as a given plane perpendicular to the a;-axis. In this 

 case it is evident that = tan 9, and therefore 



JJQ 

 --^ \/sin2^.tan2^-«2; 



whence, effecting the integration, 



F(^, «) = lo- f- ■ ^sin'* &- «^ cos^ 6' + 1 + cos^ ^ 1 



cos 6 



a . _ r u l + cos^^l 



+ 2'^'' Lv'^^^T4.' sin^^ J' 



, • _ir "■ l + cos^q 



and differentiating. 



Consequently it follows from (29), that every surface which 

 attracts in the same manner as the given plane will be represented 

 by an equation which results from the elimination of « between 

 the equations 



„ 1. r« i/sin'i^-a-'cosS^-f-l + cos^^l , .. . .,, . 2(«2 + 2) 



40. In order to examine the nature of the characteristic repre- 

 sented by these equations, wc will give to m, for the moment, 



(32) 



