[From the Proceedingt of the Cambridge Philosophical Society, Vol. n, 1876.] 



LII. On the Proof of the Equations of Motion of a Connected System. 



To deduce from the known motions of a system the forces which act on it 

 is the primary aim of the science of Dynamics. The calculation of the motion 

 when the forces are known, though a more difficult operation, is not so impor- 

 tant, nor so capable of application to the analytical method of physical science. 



The expressions for the forces which act on the system in terms of the 

 motion of the system were first given by Lagrange in the fourth section <>f 

 the second part of his Mdcanique Analytique. Lagrange's investigation may be 

 regarded from a mathematical point of view as a method of reducing the dy- 

 namical equations, of which there are originally three for every particle of the 

 system, to a number equal to that of the degrees of freedom of the system. 

 In other words it is a method of eliminating certain quantities called reactions 

 from the equations. 



The aim of Lagrange was, as he tells us himself, to bring dynamics under 

 the power of the calculus, and therefore he had to express dynamical relations in 

 terms of the corresponding relations of numerical quantities. 



In the present day it is necessary for physical inquirers to obtain clear ideas 

 in dynamics that they may be able to study dynamical theories of the physical 

 sciences. We must therefore avail ourselves of the labours of the mathematician. 

 and selecting from his symbols those which correspond to conceivable physical 

 quantities, we must retranslate them into the language of dynamics. 



In this way our words will call up the mental image, not of certain 

 operations of the calculus, but of certain characteristics of the motion of bod 



The nomenclature of dynamics has been greatly developed by those who in 

 recent times have expounded the doctrine of the Conservation of Energy, and 

 it will be seen that most of the following statement is suggested by the inv 



