CONSERVATION OF+EXTENSION-IN-PHASE 



where C represents the same value as in the preceding 

 formula, viz., the constant value of the maximum coefficient 

 of probability, and F n is a quadratic function of the differences 

 Pi ~ p i"> <?i" - Ci", etc., the phase (P x ", QJ' etc.) being that 

 which at the time t" corresponds to the phase (P/, #/, etc.) 

 at the tune t'. 



Now we have necessarily 



J*. . . 



&>i" . . . d" = 1, (53) 



when the integration is extended over all possible phases. 

 It will be allowable to set oo for the limits of all the coor- 

 dinates and momenta, not because these values represent the 

 actual limits of possible phases, but because the portions of 

 the integrals lying outside of the limits of all possible phases 

 will have sensibly the value zero. With oo for limits, the 

 equation gives 



l, (64) 



Vf Vf" 



where/' is the discriminant * of F 1 , and/" that of F". This 

 discriminant is therefore constant in time, and like C an abso- 

 lute invariant hi respect to the system of coordinates which 

 may be employed. In dimensions, like (7 2 , it is the reciprocal 

 of the 2nth power of the product of energy and time. 



Let us see precisely how the functions F' and F' f are related. 

 The principle of the conservation of the probability-coefficient 

 requires that any values of the coordinates and momenta at the 

 time t f shall give the function F' the same value as the corre- 

 _ sponding coordinates and momenta at the time t n give to F". 

 Therefore F n may be derived from F' by substituting for 

 Pi* - 9.n their values in terms of p^', . . . <?/'. Now we 

 have approximately 



* This term is used to denote the determinant having for elements on the 

 principal diagonal the coefficients of the squares in the quadratic function 

 F', and for its other elements the halves of the coefficients of the products 

 inF'. 



