Il8 PARTICULAR CASES OF EQUILIBRIUM. 



that of the corresponding asymptote is 



m 



m m' 



As the angle a cannot become greater than TT, there can only be 

 unlimited lines of force for values of smaller than 



The line of force AP l corresponding to this value separates 

 the m-m' lines of force proceeding from A, and which are unlimited, 

 from the m' which are finite and are absorbed at A'. The equation 

 of this limiting line of force is 



ma) - m'<D f = m0 = (m m') TT, or if o/ = -(TT o>). 



M 



This equation is satisfied, for O> = TT and o>' = 7r; hence the line 

 meets the axis on the left of the point A', and the point of meeting 

 O' is symmetrical with the centre of gravity of the system in refer- 

 ence to the axis AA'. We have, in fact, for any point P x of the 

 curve 



. sin (TT-O)) 



r sin to sin (TT - o> ) m 



r sin sin (TT - <o) sin (TT 

 If the angle TT - w approaches zero, we get 



limY \= , or mxO'A' = m' 

 \ r'/ m' 



As the centre of gravity O of the two masses is determined by 

 the condition m x OA = m' x O A', it follows that OA = O'A'. 

 In the case of m = 2m' (Fig. 29) we have 



7T 



= , and 2w o>' = TT, 



from which we get 



2 sin 

 2 



