8 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



and reactions of the system. The formula (6) expresses a criterion 

 of this kind in the most simple and direct manner. If we regard 

 a force as a tendency to increase a quantity expressed by x t the 

 product of the force by Sx is the natural measure of the extent to 

 which this tendency is satisfied by an arbitrary variation of the 

 accelerations. The principle expressed by the formula may not be 

 very accurately designated by the words virtual velocities, but it 

 certainly does not differ from the principle of virtual velocities (in 

 the stricter sense of the term), more than this differs from that of 

 virtual displacements, a difference so slight that the distinction of 

 the names is rarely insisted upon, and that it is often very difficult 

 to tell which form of the principle is especially intended, even when 

 the principle is enunciated or discussed somewhat at length. 



But, although the formulae (6) and (9) differ so little from the 

 ordinary formulae, they not only have a marked advantage in respect 

 of precision and accuracy, but also may be more satisfactory to the 

 mind, in that the changes considered (to which S relates), are not so 

 violently opposed to all the possibilities of the case as are those which 

 are represented by the variations of the coordinates.* Moreover, as 

 we shall see, they naturally lead to various important laws of motion. 



Transformation of the New Formula. 



Let us now consider some of the transformations of which our 

 general formula (6) is capable. If we separate the terms containing 



* It may have seemed to some readers of the Mfoanique Analytique a work of which 

 the unity of method is one of the most striking characteristics, and that to which its 

 universally recognized artistic merit is in great measure due that the treatment of 

 dynamical problems in that work is not entirely analogous to the treatment of statical 

 problems. The statical question, whether a system will remain in equilibrium in a 

 given configuration, is determined by Lagrange by considering all possible motions of 

 the system and inquiring whether there is any reason why the system should take any 

 one of them. A similar method in dynamics would be based upon a comparison of a 

 proposed motion with all other motions of which the system is capable without violating 

 its kinematical conditions. Instead of this, Lagrange virtually reduces the dynamical 

 problem to a statical one, and considers, not the possible variations of the proposed 

 motion, but the motions which would be possible if the system were at rest. This 

 reduction of a given problem to a simpler one, which has already been solved, is a 

 method which has its advantages, but it is not the characteristic method of the 

 Mtcanique Analytique. That which most distinguishes the plan of this treatise from 

 the usual type is the direct application of the general principle to each particular case. 



The point is perhaps of small moment, and may be differently regarded by others, 

 but it is mentioned here because it was a feeling of this kind (whether justified or not) 

 and the desire to express the formula of motion by means of a maximum or minimum 

 condition, in which the conditions under which the maximum or minimum subsists 

 should be such as the problem naturally affords (Gauss's principle of least constraint 

 being at the" time unknown to the present writer, and the conditions under which the 

 minimum subsists in the principle of least action being such that that is hardly satis- 

 factory as a fundamental principle), which led to the formulae proposed in this paper. 



