21-23] Statical Systems 25 



The values of the ratios 



B/UL B/jL SfjL 



OVi u(/2 OU n 



determine the direction of the linear series through the configuration 

 i, 2) ... in our diagram. At points such as Q in figs. 4 (i) and 4 (iii) 

 one or more of these ratios must become indeterminate, so that we must 

 have (say) 8/jL/S0 l = 0/0. At a point such as Q in fig. 4 (ii) we must have 

 (say) S/JL/&0! 0. Thus the three points Q in figure 4 are all determined by 

 the single condition 



A = () .................................... (14). 



23. We must now try to connect this up with the analytical condition 

 for a change of stability occurring at the configuration lf 2 , ... n and 

 the value //. of the parameter. Keeping JJL constant, the change of potential 

 energy corresponding to changes B0 lt 80 2 , ... in the values of 1} .. ..., will, 

 as in equation (10) be given by 



,,+ .................. (15) 



in which no terms of degree beyond the second need be written down when 

 80 lt B0 2 ... are supposed small. 



Let the coordinates B0 lt B0 2 ... in this quadratic expression for BW be 

 changed by a linear transformation to new coordinates fa, fa..., such that 

 B W becomes a sum of squares, say 



and let the modulus of transformation be X. 



Since the discriminant remains invariant through all linear transfor- 

 mations, we have 



b l} 0, 0, ... 

 , 6 2 , 0, . 



r W 



12> " 1 



or bj}, ... b n = \k (17). 



The condition that the configuration (17) shall be stable is that BW shall 

 be positive for all values of B0 lt S0 2 , ... B0 n , or again that expression (16) 

 shall be positive for all values of fa, fa, ...fa. This condition is that 

 b lt 6 2 , ... b n shall all be positive. 



The coefficients b lt 6 a , ... b n are called by Poincare " coefficients of stability." 

 A change from stability to instability occurs when any one of these coefficients 

 vanishes, and the values of /u- for which this occurs are, from equation (17), 

 given by 



A = (18). 



