158 PARTICULAR CASES OF EQUILIBRIUM. 



We can determine the direction of the tangent as above (145), 

 which gives 



, . sin/3 /' m'r 2 i r 2 



/== 



170. When the tangent is horizontal, we have 



or 



sin /3 = sin <o, 

 sin f}' = sin a/. 



Equation (32) becomes then 



sin (air 2 



sin u>' & r' 2 

 or replacing the sines by the opposite sides, 



r f _ i r 2 



r~~&'7*' 



which equation may be thus written 



The locus of the points where the tangent is horizontal, is there- 

 fore a sphere comprising the point A'. The centre and the radius 

 of this sphere may be calculated by formulas (30) in which k is 

 replaced by $. 



171. When the tangent is vertical, we have 



or 



sin /? = cos a). 

 sin 3' = cos /. 



