150 

 and 



Mr. S. H. Burbury on the 



«s 





2Tn _ T 

 H 2 ' 



because 



n 



a l D L1 + b 12 D l2 + • • -=2 ■ D * 



(7) We have assumed that the quadratic function 



S = a^ + b 12 x { x 2 + & c « 



must have its coefficients so chosen as that it shall be positive 

 for all values of the variables x x . . . # m . The condition that 

 this may be the case is that the determinant 



2a x bi2 . . • b, 

 2a 2 . 



D = 



In 



b\2 



2a n 



and all its coaxial minors shall be positive. Hence each ol 

 the coefficients a is positive, and ia a q —b 2 is positive for 

 all values of p and g. 



If S be the energy of an association, the above condition is 

 necessary for its stability. So for two associations near to one 

 another, let the energy be a Y Xi + b 12 x x x 2 + . . . -f a{x{ 2 + &c, 

 and let A be the double determinant X + 2a lt 2a 2 . . . 2a/. . . . 

 Then in order that the two associations, considered as a single 

 system, may be stable, A and all its coaxial minors must be 

 positive. If the expression for the energy contains no terms 

 of the form bxx f , where x belongs to one and x' to the other 

 association, A is the product of the two single determinants 

 £ + 2%, 2a 2 . . . and X±2a/, 2a 2 . . . , and if the condition be 

 satisfied for them, it is satisfied for A. But if any terms of 

 the form bxx' make their appearance, they may make A or 

 some of its coaxial minors negative, and the two associations, 

 though separately stablo, may be unstable together, and dis- 

 sociation may ensue. 



Pakt II. 



(8) Now let us suppose that S is proportional to the kinetic 

 energy, or to a part of the energy, of an association. Let the 

 number of associations, or, as we will now call them, systems, 

 be very great. Each has its own S. And we will effect a 



