168 Dadourian — Progressive Development of Mechanics. 



2G, + R« = o , 

 or G = - R a 



= R , (IV) 



where G denotes the resultant torque. By comparing the 

 expression for the moments of the kinetic reactions of the 

 elements of a rotating body with equation (IV) it may be 

 shown that the foregoing experimental definition of moment 

 of inertia is equivalent to the common analytical definition. 



In this treatment of the science of mechanics the principles 

 of the conservation of dynamical energy, of linear momentum 

 and of angular momentum are derived from the action princi- 

 ple and are used as supplementary principles. 



The form of the statement of the action principle and the 

 way in which its significance is developed do away with the 

 difficulties which arise from confounding the linear kinetic 

 reaction with a force and the angular kinetic reaction with a 

 torque. A clear-cut distinction is made between force and 

 torque on the one hand and the two forms of kinetic reaction 

 on the other. The former represent interactions oetioeen 

 matter and matter while the latter represent interactions 

 between ether and matter. On this view inertia is not a 

 resistance which bodies offer to an accelerating force. The 

 conception of two gravitating particles attracting each other 

 and at the same time resisting attraction, imparting accele- 

 ration yet offering resistance to acceleration, is obtained by 

 analogy from tug-of-war. But the analogy does not hold 

 good unless something is introduced into the conception of the 

 gravitating particles which will play the same roll as the com- 

 mon ground on which the men playing tug-of-war stand. 

 Without such a common ground there can be no tug-of-war. 

 "We can not " without destroying the clearness of our concep- 

 tions take the effect in inertia twice into account — first as mass 

 and secondly as force." The point of view outlined here com- 

 pletes the analogy by introducing the necessary " common 

 ground." 



In addition to clarifying the fundamental principle, the 

 development outlined in this paper unifies the presentation of 

 the science of Mechanics to a great extent, makes it progressive 

 and graded. A tremendous gain is thus made in economy of 

 effort. 



The following derivation of Newton's third law of motion 

 illustrates the simplicity with which dynamical laws and 

 theorems can be derived from the action principle. 



Consider two interacting particles : Let m 1 and m 2 be the 

 masses, and v, and v„ the accelerations of the particles. Fur- 



