684 



SCIENCE 



[N. 8. Vol. XLI. No. 1062 



whole problem is very complicated, and it is 

 the -writer's purpose merely to call attention to 

 the importance of surface conditions in the 

 production of the rare gases. 



Egbert H. Goddaed 

 Clabk College 



the fundamental equation of mechanics 



Me. Kent, in his recent communication, 

 invites expressions of opinion from Professor 

 Huntington and myself regarding his method 

 of explaining the principles of dynamics. My 

 own view is that Mr. Kent's explanation of the 

 effect of a constant force in giving motion to a 

 free body initially at rest is entirely sound. It 

 is, in fact, substantially the explanation I have 

 long used in the classroom as a first step in 

 establishing the fundamental equation of mo- 

 tion. Perhaps it is permissible to quote from 

 my text-book on " Theoretical Mechanics," first 

 published fifteen years ago: 



If a force of constant magnitude and direction 

 acts, for a certain interval of time, upon a body 

 initially at rest, the body will have at the end of 

 the interval a velocity whose direction is that of 

 the force, and whose magnitude is proportional 

 directly to the force and to the duration of the 

 interval, and inversely to the mass of the body. 



Since mass has already been defined as 

 quantity of matter, this statement is seen to 

 be identical in meaning with Mr. Kent's state- 

 ment that " the velocity varies directly as the 

 time and as the force, and inversely as the 

 quantity of matter." 



Mr. Kent's equation V = EFT/W is en- 

 tirely satisfactory and sufficient so long as our 

 study is confined to the case in which a force 

 whose direction and magnitude remain con- 

 stant acts upon a body otherwise free and 

 initially at rest. This is, however, a very ex- 

 ceptional case. The fundamental principle in 

 its generality can be expressed only by intro- 

 ducing the notion of instantaneous rate of 

 change of velocity, i. e., acceleration. When 

 this is done Mr. Kent's statement quoted 

 above must be replaced by the statement that 

 "the acceleration varies directly as the force 

 and inversely as the quantity of matter," 

 while his equation V = KFT/W is superseded 

 by the more general one a = KF/W. This is 



identical with equation (5) of my former com- 

 munication,! except that quantity of matter is 

 there represented by m instead of W. 

 To pass from the equation 



acceleration = X X q^^^ti^ of °^^tter 



force *■ ' 



to the equation 



, ,. quantity of matter 



acceleration = ^ ^ f 2) 



force 



of course requires that units should be defined 

 so that unit force acting on unit quantity of 

 matter causes unit acceleration. Mr. Kent 

 regards this as an objection to equation (2), 

 If the objection is valid a similar one seems to 

 apply to his own procedure. His equation 

 FT 



F = 32.1740 



W 



is true only because his unit force is defined 

 as the force which would give a pound of 

 matter an acceleration of 32.1740 ft./sec.* 

 The statement that the accurate value 

 Z = 32.1740 is found as the result of "the 

 most refined experiments, involving precise 

 measurements of both F and W, and of 8, the 

 distance traversed during the time T, from 

 which V is determined " is quite misleading. 

 The stated value of Z^ is not based upon any 

 refined measurements of the character de- 

 scribed, but upon a purely ideal definition of 

 the unit force ; just as the value K^=l results 

 from a different ideal definition. 



If there is any reason for preferring the 

 set of units which makes Z'::= 32.1740 to that 

 which makes jS^ = 1 in equation (1), it is not 

 because the former is any more easily under- 

 stood than the latter. " The force which, act- 

 ing upon a pound of matter, would cause an 

 acceleration of 32.1740 ft./sec.^ " is the same 

 kind of a definition as " the force which, act- 

 ing upon a pound of matter, would cause an 

 acceleration of 1 ft./sec.^ " It is true that the 

 former of the two units of force thus defined 



1 Science, April 23, 1915, p. 609. It is well 

 known that Mr. Kent objects to the use of the word 

 mass for quantity of matter; my present object is 

 to make my meaning clear rather than to invite an 

 unprofitable discussion over a purely verbal ques- 

 tion. 



