Dr. Hirst on Equally Attracting Bodies. 171 



From this expression we deduce at once the conditions of coin- 

 cidence and orthogonality of the corresponding normal vector- 

 planes. They will coincide when v=0 or =7r, i. e. when 



du dui du dui 



ddW~d^"d^' 

 or 



du duj _ du dui 



dd''d0~d^'d^' ^^' 



and they will be perpendicular to each other when v= ^, i. e. 

 when du duy 1 dudu^_ 



de'do'^^^'d^Ti^ ' • • ^ > 



It will also be remarked that this equation (10), identically ful- 

 filled, expresses the condition of orthogonality between two 

 systems of cones having the origin for common vertex, and the 

 equations u = c, Mj = Ci; where c and c, are variable parameters. 

 12. Having thus established the necessary analytical formulae, 

 we proceed next to examine the mutual properties of any two 

 equally attracting surfaces {u) and (mj). According to the 

 theorem of art. 3 and the equation (6), the necessary and suffi- 

 cient condition to be fulfilled, identically, by two such surfaces is 



/duy _1_/'^V_ /"^iV 1 {du^Y 



\dd) "^ sin^e\d<}>) ~\de) "^ siu2^w0/ • • ^ ^ 



As remarked in art. 6, a glance at this equation is sufficient to 



show that it is satisfied when «=+«,, or log-= +log— , that 



c e, 



when 



- = — , and when - = — 

 c c^ c r. 



'1 " '1 



in the first case the surfaces are similar, in the second inverse to 

 each other ; and in both cases, as may be seen from equation (9), 

 all corresponding normal vector-planes coincide. It is also ma- 

 nifest, conversely, that this coincidence only exists when the 

 equally attracting surfaces are similar or inverse to each other, 

 for the equations (9) and (11) can only be satisfied, simulta- 

 neously, when M= +M,. 



13. The inquiry here naturally presents itself — under what 

 conditions will two equally attracting surfaces have all their cor- 

 responding normal vector-planes perpendicular to each other ? 

 In this case it is the equations (10) and (11) which, by hypo- 

 thesis, are simultaneously and identically fulfilled. By means 

 of the substitution 



log (tan ^^)=ft), 



