157-159] VARIATIONAL EQUATION OF MOTION. 141 



at any time. The general equation above written is therefore equivalent to 

 all the equations of motion of the system. It is called the general variational 

 equation of motion. 



The expression of the coefficients of 80, d&amp;lt;p,... in the left-hand member is 

 beyond the scope of this book. 



CRITICAL NOTE. 



THE conception of bodies, as continuous and made up of particles, and the 

 conception of the mutual actions of bodies, as made up of forces between 

 particles, are, as a matter of historical fact, the two conceptions upon which 

 the existing science of Mechanics has been based. They possess further the 

 advantages, (1) that it is possible to found upon them a strictly logical 

 deductive theory, in fact the theory sketched in the preceding Chapters and 

 to be exemplified in subsequent Chapters, and (2) that this theory provides 

 an adequate abstract formulation of the rules obeyed by the motions of the 

 bodies of the solar system, and of matter in bulk under ordinary conditions. 

 It has thus historically developed into a scheme which successfully coordinates 

 an immense number of disconnected observations concerning matters of fact. 

 Accordingly this theory constitutes a science, a logically valid and practically 

 valuable method of representing observed facts by abstract formulas. 



An objection has been raised against this method of formulation*, that at 

 the outset it admits a possibility which it afterwards excludes the possibility 

 of non-conservative positional forces. This objection seems to me to have no 

 weight, since we should expect that, in a theory logically deduced from 

 definitions and postulates, the postulates could not all be introduced at the 

 beginning, but rather that, at certain stages of the process of deductive 

 argument, it would happen that a choice among different possibilities would 

 offer itself. At such stages additional postulates might be introduced, and, 

 in fact, the postulating of the Law of Gravitation as a general law of force is 

 an example of the method of making such a choice among postulates. In all 

 such cases that postulate is to be preferred which accords with the simplest 

 expression of observed facts. 



There are however other objections of a more serious character. The first 

 of these is that the theory cannot be complete, or that it aims at an unattain 

 able precision. In order to state completely in terms of the theory the rules 

 that govern the motions of bodies we should require to know what law of 

 force between the hypothetical particles ought to be assumed in the cases in 



* By Hertz in his Principien der Mechanik, Leipzig, 1894. 



