OF THE ENERGIES OF A SYSTEM. 99 



Similar formulae may be used to derive V q or <j> q for the 

 compound system, when one of these quantities is known. 

 as function of the potential energy in each of the systems 

 combined. 



The operation represented by such an equation as 



C 



= I 



01 02 



e e 



is identical with one of the fundamental operations of the 

 theory of errors, viz., that of finding the probability of an error 

 from the probabilities of partial errors of which it is made up. 

 It admits a simple geometrical illustration. 



We may take a horizontal line as an axis of abscissas, and lay 

 off 61 as an abscissa measured to the right of any origin, and 

 erect e^i as a corresponding ordinate, thus determining a certain 

 curve. Again, taking a different origin, we may lay off e 2 as 

 abscissas measured to the left, and determine a second curve by 

 erecting the ordinates e^. We may suppose the distance be- 

 tween the origins to be e 12 , the second origin being to the right 

 if e 12 is positive. We may determine a third curve by erecting 

 at every point in the line (between the least values of ei and e 2 ) 

 an ordinate which represents the product of the two ordinates 

 belonging to the curves already described- The area between 

 this third curve and the axis of abscissas will represent the value 

 of e^ 12 . To get the value of this quantity for varying values 

 of 6 12 , we may suppose the first two curves to be rigidly con- 

 structed, and to be capable of being moved independently. We 

 may increase or diminish e 12 by moving one of these curves to 

 the right or left. The third curve must be constructed anew 

 for each different value of e 12 . 



