32 General Dynamical Principles [CH. n 



Using the values of T R , U and / given by equations (22) to (24), 

 27 T, = [2m, (^ 2 + 2 



This expression, being a sum of squares, is always positive. Thus, since / is 

 necessarily positive and independent of sb, y, z, it appears that T 8 is always 

 positive and is quadratic in x, y, z. 



The equation of energy, T + W = cons, now assumes the form 



M 2 

 T 8 + TF+=cons ............................ (43). 



This is of the same form as equation (42), T s replacing T R and W + M 2 /2/ 

 replacing W - ^w 2 /. By the argument already used in 28, it now appears 

 that configurations for which 



.................................... (44) 



is an absolute minimum (M being kept constant) will be thoroughly stable, 

 while configurations for which this expression is not an absolute minimum 

 will be secularly unstable, and may or may not be ordinarily unstable. 



30. As we pass along a linear series of configurations of equilibrium of 

 a rotating system, starting from a part of the series which is known to be 

 stable, the configurations will become secularly unstable as soon as 



W -\<*?I ( =constant) ....................... .(45) 



or W +M 2 //(M = constant) ........................ (46) 



ceases to be ah absolute minimum, the former expression referring to a 

 problem in which the mass is compelled by external forces to rotate at a 

 constant rate &>, while the latter refers to a problem in which the mass is 

 rotating freely in space. 



It is now clear that the theory of linear series and stability developed in 

 1820 will be exactly applicable to the problem of the secular stability of 

 a rotating mass, W being replaced in the argument of those sections by the 

 appropriate one of expressions (45) or (46). Secular stability is lost at a 

 " turning point " or " point of bifurcation." At a turning point stability is 

 lost entirely ; at a point of bifurcation it may be lost or may be transferred 

 to the branch series through the point according as the branch series turns 

 downwards or upwards in the appropriate diagram. 



