346 BELL SYSTEM TECHNICAL JOURNAL 



"clearance" {S — L) between the adjacent ends of the magnets to 

 be extremely small by comparison with 5 and L, and by considering 

 only values of 6 which are so small that R itself remains small by 

 comparison with L; otherwise the equations will be hopelessly intri- 

 cate, and they are more than bad enough even with this restriction. 

 Happily the model possesses some of the required properties even 

 when limited by this restriction. 



The torque T exerted by the field II upon either magnet is given by 



T = MIIL sin {a - e). (3) 



The general condition for equilibrium is 



r - r = 0. (4) 



The special condition for "neutral" or "labile" equilibrium, i.e. 

 for the state of incipient capsizal, is 



d{T - T')/dd = 0. (5) 



The values of // and 6, obtained by solving (4) and (5) as simul- 

 taneous equations, are the fieldstrength just sufficient to produce 

 capsizal and the angle of deflection attained just before the overturn; 

 they are obtained as functions of the variable a, and of the constants 

 M, L, S which are features of the model. 



Solving these equations, however, is easier said than done; they 

 prove surprisingly intractable. Only in one particular case is the 

 solution easy: we must choose values of a so near to 90°, and suppose 

 the clearance and consequently the deflections so small, that the 

 cosine of (a — 6) may be set equal to zero. In this case equation (5) 

 is reduced to the form 



dT'ldd = const, ^(sin dlR')/dd = 0, (6) 



which, if we write a for S/L, is found equivalent to 



(a2 + 1 - 2a cos d) cos 6 = 3a sin^ 6. (7) 



Putting a = 1 + e — so that e stands for the quantity (S — L)/L, 

 which by hypothesis is small — and neglecting powers of e higher than 

 the second, we arrive at the equations: 



cos d, = l- h-; sin e, = e/ ^|2 = {S - L)IL ^[2, (8) 



for the value dc of the deflection just at the verge of capsizal; and 

 putting this into equation (4), we get 



//, = -^ . (9) 



