736 SCIENCE AND HYPOTHESIS 



position to another, there is one and only one for which the mean 

 action is a minimum. The principle of least action is therefore suffi- 

 cient for the determination of the differential equations which define 

 the variations of the parameters q. The equations thus obtained are 

 another form of Lagrange's equations. 



To form these equations we need not know the relations which 

 connect the parameters g with the co-ordinates of the hypothetical 

 molecules, nor the masses of the molecules, nor the expression of U 

 as a function of the co-ordinates of these molecules. All we need 

 know is the expression of TJ as a function of the parameters q, and 

 that of T as a function of the parameters q and their derivatives 

 i.e., the expressions of the kinetic and potential energy in terms of 

 experimental data. 



One of two things must now happen. Either for a convenient choice 

 of T and U the Lagrangian equations, constructed as we have indi- 

 cated, will be identical with the differential equations deduced from 

 experiment, or there will be no functions T and U for which this 

 identity takes place. In the latter case it is clear that no mechanical 

 explanation is possible. The necessary condition for a mechanical 

 explanation to be possible is therefore this: that we may choose the 

 functions T and U so as to satisfy the principle of least action, and 

 of the conservation of energy. Besides, this condition is sufficient. 

 Suppose, in fact, that we have found a function U of the parameters 

 q, which represents one of the parts of energy, and that the part of 

 the energy which we represent by T is a function of the parameters q 

 and their derivatives; that it is a polynomial of the second degree 

 with respect to its derivatives, and finally that the Lagrangian equa- 

 tions formed by the aid of these two functions T and U are in con- 

 formity with the data of the experiment. How can we deduce from 

 this a mechanical explanation? U must be regarded as the potential 

 energy of a system of which T is the kinetic energy. There is no 

 difficulty as far as U is concerned, but can T be regarded as the vis 

 viva of a material system? 



It is easily shown that this is always possible, and in an unlimited 

 number of ways. I will be content with referring the reader to the 

 pages of the preface of my Electricite et Optique for further details. 

 Thus, if the principle of least action cannot be satisfied, no mechanical 

 explanation is possible; if it can be satisfied, there is not only one 

 explanation, but an unlimited number, whence it follows that since 

 there is one there must be an unlimited number. 



One more remark. Among the quantities that may be reached by 

 experiment directly we shall consider some as the co-ordinates of our 

 hypothetical molecules, some will be our parameters q, and the rest 

 will be regarded as dependent not only on the co-ordinates but on 

 the velocities or what comet? to the same thing, we look on them 



