12 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



The first part is evidently constant with reference to variations of 

 x\ y', z\ and may, therefore, be neglected. With respect to the second 

 part we observe that by the general formula of the motion we have 



for all values of Sx, 8y, Sz which are possible and reversible before the 

 addition of the new constraints. But values proportional to x\ y', z', 

 and of the same sign, are evidently consistent with the original con- 

 straints, and when the components of acceleration are altered to 

 x+x, y + y', z + z', variations of these quantities proportional to and 

 of the same sign as x', y', z' are evidently consistent with the 

 original constraints. Now if these latter variations were not possible 

 before the accelerations were modified by the addition of the new forces 

 and constraints, it must be that some constraint was then operative 

 which afterwards ceased to be so. The expression (22) will, therefore, 

 be equal to zero, provided only that all the constraints which were 

 operative before the addition of the new forces and constraints, remain 

 operative afterwards.* With this limitation, therefore, the expression 

 (23) must have the greatest value consistent with the constraints. 

 This principle may be expressed without reference to rectangular 

 coordinates. If we write u r for the relative acceleration due to the 

 additional forces and constraints, we have 



and expression (23) reduces to 



2(Z'tf+ Y'y'+Z'z'-^mu'*). (24) 



If the sum of the moments of the additional forces which are 

 considered is represented by (Qdq) (the q representing quantities 

 determined by the configuration of the system), we have 



We may distinguish the values of -^f immediately before and imme- 



diately after the application of the additional forces and constraints 

 by the expressions q and q + q'. With this understanding, we have, 

 by differentiation of the preceding equation, 



* As an illustration of the significance of this limitation, we may consider the con- 

 dition afforded by the impenetrability of two bodies in contact. Let us suppose that if 

 subject only to the original forces and constraints they would continue in contact, but 

 that, under the influence of the additional forces'and constraints, the contact will cease. 

 The impenetrability of the bodies then ceases to be operative as a constraint. Such 

 cases form an exception to the principle which is to be established. But there are no 

 exceptions when all the original constraints are expressed by equations. 



