



ENSEMBLE OF SYSTEMS. 51 



linear functions of the JD'S.* The coefficients in these linear 

 functions, like those in the quadratic function, must be regarded 

 in the general case as functions of the <?'s. Let 



2e p = < 2 + w 2 2 ... + iv 2 (133) 



where MJ . . . u n are such linear functions of the p'a. If we 

 write 



for the Jacobian or determinant of the differential coefficients 

 of the form dp/du, we may substitute 





for dp 1 . . . dp n 



under the multiple integral sign in any of our formulae. It 

 will be observed that this determinant is function of the <?'s 

 alone. The sign of such a determinant depends on the rela- 

 tive order of the variables in the numerator and denominator. 

 But since the suffixes of the it's are only used to distinguish 

 these functions from one another, and no especial relation is 

 supposed between a p and a u which have the same suffix, we 

 may evidently, without loss of generality, suppose the suffixes 

 so applied that the determinant is positive. 



Since the w's are linear functions of the />'s, when the in- 

 tegrations are to cover all values of the jt?'s (for constant #'s) 

 once and only once, they must cover all values of the w's once 

 and only once, and the limits will be oo for all the u's. 

 Without the supposition of the last paragraph the upper limits 

 would not always be + oo , as is evident on considering the 

 effect of changing the sign of a u. But with the supposition 

 which we have made (that the determinant is always positive) 

 we may make the upper limits + oo and the lower oo for all 

 the t*'s. Analogous considerations will apply where the in- 

 tegrations do not cover all values of the p's and therefore of 



* The reduction requires only the repeated application of the process of 

 'completing the square* used in the solution of quadratic equations. 



