80 The Law of Distribution [CH. v 



90. We may express the energy E of the system in the forms 



E = V+L ................................................ (193) 



= V + HciV + c 2 7 ?2 2 +...+c, l 77 n 2 ) ............... (194), 



where V is the potential energy, a function of the q's only, and c 1} c 2 ... are 

 also functions of the q's only. 



The volume of the generalised space for which qi,q Z '--q n lie within 

 specified ranges dq l} dq 2 ... dq n , while rj l , r) 2 ... rj n have all values such that 

 E < E is given by 



dq t dq 2 ... dq n I It . . .d^ , drj 2 ... drj n .................. (195), 



where ij 1 , rj 2 ... rj n have all values subject to 



GI ^ 2 + c 2 77 2 2 +... + c n r /n *< 2(^-7) ............... (196). 



The integral is a Dirichlet Integral* of which the value is known to be 



-27)* ............... (197). 



Differentiating with respect to E , we find that the volume representing 

 systems for which q lt q 2 ...q n lie within the same range as before, while E 

 lies between E and E + dE, is 



n 



l (198). 



r ; 



If we introduce a new condition that r) n is to lie between ij n and v) n + drj n , 

 the alterations necessary to transform expression (198) to suit the new con- 

 ditions will consist in writing n 1 for n, 2E 27 c n t} n z for 2^ 27 

 and introducing the new differential dr) n . Making these alterations the 

 expression becomes 



...(199). 



ordinates. Unfortunately it is not always possible to find coordinates satisfying these conditions. 

 To take the simplest case, the kinetic energy of rotation of a rigid body can be expressed as a 

 sum of squares in many ways, but in no case are the coordinates true Lagrangian coordinates. 

 If, for instance, we write 



2L = . 



we know that I o^dt, etc., are not true Lagrangian coordinates. 

 * Williamson, Integral Calculus, p. 320. 



