64 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



Principle. Lagrange made it the basis of the entire subject of 

 dynamics. 1 ) We may interpret 18) in terms of the principle of 

 virtual work by means of the introduction of the conception of 

 effective forces due to d'Alembert. 



If a system of particles is not free, when acted on by certain 

 impressed forces it will not take on the same motion as if there 

 were no constraint, the reactions causing it to deviate from this 

 natural motion. Having found the actual motion, we know the 

 system of forces that would produce it, if there were no constraints. 

 These are termed the effective forces and if we represent them by 

 X r ', Y r ', Z/, they are given by the equations 



d* x d z y d^ z 



X r ' = m r ^, Y r ' = m r ^f, Z r ' = m r -^- 



The equation 18) accordingly states that the reversed effective forces, 

 - X', Y', Z' together with the impressed forces, X, Y, Z, will 

 form a system in equilibrium. 



We may regard the principle from another point of view. 

 When a body is set in motion with an acceleration, it reacts on the 

 agent which produces the motion, and this kinetic reaction has the 

 properties of any force whatsoever. For instance if the accelerating 

 agency is due to contact with a second moving body, the second 

 body is retarded by a force, and this force is the reaction of the 

 first. This kinetic reaction is measured by the components 



and is thus in the opposite direction to the acceleration experienced 

 by the body. The reaction is often termed the Force of Inertia, a 

 very expressive term, representing in tangible form the fundamental 

 property of inertia, possessed by all matter, this property being that 

 matter reacts against, or in ordinary language resists, being put in 

 motion. (By the use of the term resists we in no wise mean 

 prevention of motion - - the use of the term has been objected to, 

 and Maxwell 2 ) has jokingly remarked that we might as well say that 

 a cup of tea resists being sweetened, because it does not become 

 sweet until we add sugar. The meaning here is precisely similar - 

 we mean that matter does not move until it is moved by some agent 

 external to itself. It is hardly likely that confusion can be caused 

 by the use of such common phrases, which indeed seem to attribute 

 volition to matter - - we shall accordingly make no attempt to avoid 

 them.) We may thus define matter as that which can exert forces of 



1) Lagrange, Mecanique Analytique, 1. 1, p. 267. The equation 18) although 

 first explicitly given by Lagrange, will be referred as "d'Alembert's equation", 

 as briefer than u Lagrange's equation of d'Alembert's Principle". 



2) Maxwell, Scientific Papers, Vol. II, p. 779. 



