Apbil 23, 1915] 



SCIENCE 



609 



(d) If bodies whose masses are m, m' are 

 acted upon, hy equal forces, causing accelera- 

 tions a, a', then 



i = ^ . (3) 



a m 



(e) If hodies whose masses are m,, m' are 

 acted upon hy forces F, F' such that equal 

 accelerations are caused, then 



(4) 



Equations (2), (3) and (4) are all particular 

 cases of (1), but it requires two of them to 

 express the whole import of (1), and there is 

 no reason for regarding (2) as more funda- 

 mental or more important than (3) or (4). 

 Any single equation which may properly be 

 called the fundamental equation of dynamics 

 must be equivalent in import to equation (1). 



This does not mean that it is never allow- 

 able or advantageous to use the less general 

 equations ; on the contrary, problems and illus- 

 trations falling under these special cases are 

 undoubtedly helpful to students. But the ob- 

 ject should be to lead up to an understanding 

 of the fully general principle expressed above 

 in paragraphs (a) and (&) and in equation (1). 



When this principle is fully understood, 

 it is seen that equation (1) enables us to deter- 

 mine the acceleration of any body of known 

 mass due to the action of any known force as 

 soon as we know the acceleration of one body 

 of known mass due to the action of one known 

 force. For practical use it is advantageous to 

 express the general equation in a more concise 

 form. It is readily understood that (1) is 

 equivalent to the equation 



a = k-, (5) 



in 



in which A; is a constant of which the value 



depends upon the units chosen for expressing 



acceleration, force and mass; and that the 



still more concise form 



F 

 a = - (6) 



m 



results if units are so chosen that unit force 



acting upon unit mass causes unit acceleration. 



The foregoing is essentially the Newtonian 



explanation of the second law of motion as 



interpreted by Thomson and Tait and accepted 

 by other high authorities. In essential mean- 

 ing there is no difference between this and the 

 method advocated by Professor Huntington. 

 The word mass is, indeed, avoided in his state- 

 ment ; but in recognizing the importance of the 

 fact "that different bodies require different 

 amounts of force to give them any specified 

 acceleration " he recognizes the reality and 

 fundamental importance of the body-constant 

 which is usually designated as mass. By 

 whatever name this constant may be called, it 

 must play a part in the theory equivalent to 

 that taken by mass in the equations given 

 above. In Professor Huntington's presenta- 

 tion this part is taken by " standard weight," 

 defined as the force required to give the body 

 the acceleration 32.1740 ft./sec.^ This does 

 not conflict with the theory outlined above; in 

 fact since the forces required to give different 

 bodies a specified acceleration are by equation 

 (4) proportional to their masses, standard 

 weight as above defined may serve as a valid 

 measure of mass. In explaining this method, 

 however, it is important to make perfectly 

 clear the fact that the quantity called standard 

 weight is in reality the measure of a body- 

 constant and is quite independent of gravity, 

 in spite of the fact that it is given a name 

 which is almost always associated with gravity. 

 If properly safeguarded in this respect. Pro- 

 fessor Huntington's method of developing 

 fundamental principles is, I believe, logically 

 defensible. Whether it meets the needs of 

 beginners as well as that based upon the New- 

 tonian treatment of mass may, however, be 

 questioned. 



To start with the notion of mass defined 

 provisionally as " quantity of matter " has the 

 same kind of advantage as starting with the 

 " spring-balance " definition of force. Both 

 definitions have a sufficiently definite mean- 

 ing, gained from ordinary experience, to be of 

 service in a preliminary explanation of the 

 laws of motion. In comparing the masses of 

 bodies composed of one homogeneous substance 

 the significance of the words " quantity of 

 matter " is indeed readily recognized, and it is 

 distinctly helpful to generalize this notion even 



