280 Dr. Hii'st 07i Equally Attracting Bodies. 



from the last equation we deduce, immediately, 



a _ sin^ ^ . sin 20 /o-\ 



a = + VW-- '''°> 



where, for brevity, we have made 



U = (1 + cos^ e + a sin^ 6 cos 2<^f- If- sin^ 6 sin^ 2</). 

 From (35) we deduce, too, 



Vsin-* 6'-«2 cos^ e= ± — ^ [(1 -I- cos-6') cos2(/) + « sin^^], (3G) 



where the upper and lower sign umst correspond, respectively, 

 with those in (35), in order that the result of differentiating 



-^ = a according to 6, may be identical with that obtained by 



differentiating: 



= \/t 



1a~\/ tan^ 6 Atti = -■ — tt- ^~ '^^ 



"P V sm^^ sm^.cos 



according to (p. Further, by (35), 



Vh'^u^ + 4! , l + cos2 6' + fflsin'-^^cos20 ,„^. 



2 - VV 



where, again, the upper or lower sign must be employed accord- 

 ing as the upper or loM'er signs in (35) and (36) are adopted, 

 otherwise the second equation in (31) would not bo identically 

 satisfied. Substituting these values (36) and (37) in the first of 

 the equations (34), we find 



r _c' _ sin^ g[(l + cos^ 0) cos 20 + a sin^^] + (1 + cos^ 6) \/\J 



c'~r'~ cos^[l + cos2^ + «sin'^^cos20 + « -v/U 



which, by an easy but somewhat tedious calculation, is reducible 

 to either of the follovvins; forms : — 



^^£;^fl(l + cos^^) + sin^^cos20± y/U 

 c r' cos 6 



-4b^co&d 



(38) 



a(l + cos2^) + sin26'cos20+ V\j J 



where, as usual, the upper and lower signs respectively cor- 

 respond. 



43. In order clearly to distinguish between the four surfaces 

 represented in (38), kt the pair of inverse surfaces (r) and (?•') 

 correspond to the ujiper signs, and {i\) and (?■',), also inverse to 

 each other, to the lower signs. This being the case, the iden- 

 tical expressions in (38) show, at once, that (?•) and ( — r,), (r') 



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