180 



WORK 



dW 

 The positions of equilibrium are now given by - = 0, or 



-Z) =0, 

 d<p 



or, substituting for r from equation (a), 



X, X (a 2 b z ~) sin cos _ 



wa sm0 + -(a 2 & 2 ) sm0 cos0 -- ^ } 0. 



* Va 2 COS 2 _|_ J2 s i n 20 



Rationalizing, we find that roots are given by sin = 0, and also by 

 [wa + -(a 2 - 6 2 ) cos 01 2 (a 2 cos 2 + Z> 2 sin 2 0) - X 2 (a 2 - & 2 )~ cos 2 = 0, 



which reduces to 



[wa + - (a 2 - 6 2 ) cos 01 2 f(a 2 - 6 2 ) cos 2 + & 2 1 - X 2 (a 2 - & 2 ) 2 cos 2 = 0, (c) 



an equation of the fourth degree in cos 0. 



The roots of sin = are = 0, TT, so that there are always two 



positions of equilibrium at A, A', the ends of the major axis. Equation 



(c), being of the fourth degree, may have 

 0, 2, or 4 real roots in cos 0. The equa- 

 tion as it stands has been obtained by 

 squaring both sides of the equation to 

 be satisfied, and in doing this we have 

 doubled the number of roots of the true 

 equation. Thus the true equation will 

 only be satisfied by 0, 1, or 2 real roots 

 in cos 0. In other words, between A and 

 A', on either side of the wire, there can 

 be at most two positions of equilibrium. 

 It would be a tedious piece of work to find the actual values of the roots 



d 2 W 

 for cos0, and then determine the signs of the values of -- correspond - 



2 



ing to these roots. The question is, however, very much simplified by 

 using the general theory of stable and unstable configurations. 



If we put X = in expression (&), we obtain as the potential energy in 

 the case in which X is vanishingly small in comparison with w, 



W ^ 



0=0 



of which the graph is shown in fig. 100. Here there are only two positions 

 of equilibrium, namely = and = TT, the former being unstable (7) 

 and the latter stable (S). 



