716 Mr. E. Buckingham : Notes on 



Now of course equation (52) is not true, for we know that 

 equation (50; is correct; and the reason for this false result 

 is that Newton's second law is really one of the essential 

 facts and must not be ignored. It may be utilized in the 

 ordinary manner by setting [/] = [m/£~ 2 ], as was done in 

 obtaining equation (50) ; but another method may also be 

 pursued which leads to the correct result, even though an 

 independent unit of force be used. 



^Vhen the second law of motion is expressed, as is often 

 done, by saying "force equals mass times acceleration,"'* it is 

 tacitly assumed that normal units are to be employed — i. e., 

 that the unit of force is to be the force which gives unit- 

 acceleration to unit mass. But we may just as well 

 take any multiple of this as the unit ; and instead of 

 writing [/' j = \_mlt~ 2 '] as we usualjy do for conciseness, 

 we ought in general to write 



[/] = [C»»?r'] (53) 



in which C is an arbitrarily selected number, or derivation 

 constant, which characterizes the system of units and may 

 have any value de-ired, but is naturally taken to be unity, 

 as in the C.G-.S. system, when there is no reason for doing 

 otherwise. 



So lonu- as C is a fixed number, the unit of force is not 

 independent but is fixed by those of mass, length, and time 

 in accordance with equation (53) and the value selected 

 for C. But if the unit of force is made independent, this 

 converts C from a fixed number into a dimensional quantity 

 which has, by (53), the dimensions 



[C] = [fm-H-H 2 ] (54) 



in terms of the four fundamental units f } m, 1, t. And 

 while still using I fundamental units, we may utilize the 

 second law of motion by including this new quantity in 

 the list and writing, as the initial equation, 



F (R, D, S, p, ft, C) = (55) 



The solution of this by the II theorem, with equations (51) 

 and (54), is _ 



pm^ G H-frh (o6) 



and upon comparison with (50) it is seen that since <p and <j> x 

 are of unknown form, the two results are identical if is a 

 constant, i.e. as soon as any particular fixed relation between 

 the units j) m, 1. and t has been adopted. 



