ENSEMBLE OF SYSTEMS. 85 



( + D) (u - u) = 0, (256) 



where D represents the operator 2 d/d. 

 Hence 



(e + D) A (u - u) = 0, (257) 



where h is any positive whole number. It will be observed, 

 that since e is not function of , (e + D) h may be expanded by 

 the binomial theorem. Or, we may write 



(e + />) u = (e + D) u, (258) 



whence (e + X>)* u = (e + D) h u. (259) 



But the operator (e + D)*, although in some respects more 

 simple than the operator without the average sign on the e, 

 cannot be expanded by the binomial theorem, since e is a 

 function of with the external coordinates. 

 So from equation (254) we have 



< 26 ) 



whence (~ + J;)* ( - u) = ; (261) 



The binomial theorem cannot be applied to these operators. 



Again, if we now distinguish, as usual, the several external 

 coordinates by suffixes, we may apply successively to the 

 expression u u any or all of the operators 



, 



, etc. (264) 



