PROOF OF THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM. 309 



gations in Thomson and Tait's Natural Philosophy, especially the method of 

 beginning with the case of impulsive forces. 



*I have applied this method in such a way as to get rid of the explicit 

 consideration of the motion of any part of the system except the co-ordinates or 

 variables on which the motion of the whole depends. It is important to the 

 student to be able to trace the way in which the motion of each part is 

 determined by that of the variables, but I think it desirable that the final equa- 

 tions should be obtained independently of this process. That this can be done is 

 evident from the fact that the symbols by which the dependence of the motion 

 of the parts on that of the variables was expressed, are not found in the final 

 equations. 



The whole theory of the equations of motion is no doubt familiar to mathe- 

 maticians. It ought to be so, for it is the most important part of their science 

 in its application to matter. But the importance of these equations does not 

 depend on their being useful in solving problems in dynamics. A higher function 

 which they must discharge is that of presenting to the mind in the clearest 

 and most general form the fundamental principles of dynamical reasoning. 



In forming dynamical theories of the physical sciences, it has been a too 

 frequent practice to invent a particular dynamical hypothesis and then by means 

 of the equations of motion to deduce certain results. The agreement of these 

 results with real phenomena has been supposed to furnish a certain amount of 

 evidence in favour of the hypothesis. 



The true method of physical reasoning is to begin with the phenomena and 

 to deduce the forces from them by a direct application of the equations of 

 motion. The difficulty of doing so has hitherto been that we arrive, at least 

 during the first stages of the investigation, at results which are so indefinite 

 that we have no terms sufficiently general to express them without introducing 

 some notion not strictly deducible from our premisses. 



It is therefore very desirable that men of science should invent some method 

 of statement by which ideas, precise so far as they go, may be conveyed to the 

 mind, and yet sufficiently general to avoid the introduction of unwarrantable 

 details. 



For instance, such a method of statement is greatly needed in order to 

 express exactly what is known about the undulatory theory of light. 



* [In the Author's treatise On Electricity and Magnetism, Vol. II. Part IV. Chap, v., the reader 

 will find the subject treated at length from the point of view advocated in the text.] 



