ENSEMBLE OP SYSTEMS. 



The multiple integrals in 



average rallies of the expressions In the brackets, 



may therefore set equal to zero. The first gives 



as already obtained. With this relation and (191) we get 

 from the other equations 



We may add for comparison equation (205), which might be 

 derived from (236) by differentiating twice with respect to 8 : 



The two last equations give 



dl 



(A l - A l )(e - e) = (6 - )'. (245) 



e?e 



If i/r or e is known as function of 0, Oj, Oj, etc*, (e e) 2 may 

 be obtained by differentiation as function of the same variables. 

 And if i|r, or A v or 17" is known as function of 8, O 



(e e) may be obtained by differentiation. But 

 (^A l A^y- and (^A l A^) (^ 2 A 2 ) cannot be obtained in any 



similar manner. We have seen that (e e) 2 is in general a 

 vanishing quantity for very great values of TI, which we may 

 regard as contained implicitly in as a divisor. The same is 



true of (A^ A^) (e e). It does not appear that we can 



assert the same of (A-^ -4 X ) 2 or (A l A^) (^ 2 -4 2 ), since 



6 



