714 Mr. E. Buckingham : JSotes on 



quantity, just as a particular piece of metal might serve,, 

 at the same time, as a standard of mass, a standard of 

 length, a standard of thermal capacity, etc. 



13 The number of Fundamental Units required. 



The foregoing discussion leads naturally to the question: 

 How many fundamental units are really needed ? The 

 answer is evidently important, for dimensional reasoning- 

 enables us to reduce the number of independent variables to 

 be considered, by the number k of fundamental units used ; 

 and the smaller k is, the less advantage is to be derived 

 from the dimensional method and the more indeterminate is 

 the result obtained. 



It might appear, at first sight, that a new discovery 

 which enabled us to dispense with a fundamental unit 

 previously regarded as necessary, would be a disadvantage ;, 

 and that, with increasing knowledge of general physical 

 laws, the utility of the dimensional method would ulti- 

 mately evaporate without residue. Common sense tells us, 

 of course, that this appearance is fallacious ; and in con- 

 nexion with the question what would happen to our 

 dimensional reasoning if temperature could be derived 

 from mass, length, and time, Lord Rayleigh remarked: 

 ('Nature/ August 12, 1915, p. 614): "It would indeed, 

 be a paradox if the further knowledge of the nature of 

 heat afforded by molecular theory put us in a worse position 

 than before in dealing with a particular problem." But 

 while it is clear enough that new knowledge is not going 

 to be disadvantageous, it is well to examine the question 

 a little farther. 



In considering the treatment of dimensional constants, 

 whether universal or not, we saw that when k was decreased 

 by unity n had simultaneously to be decreased by. unity, 

 so that the solution did not become more indefinite, the 

 indeterminate operator in the final resuit having the same 

 number of dimensionless operands as before. The degree of 

 indefiniteness ol: the result was the same whether the 

 number of fundamental units was reduced as far as 

 possible, or not ; and to that extent the number of funda- 

 mental units used was a matter of indifference. It is easily 

 shown that an increase of the number of fundamental units 

 also leaves the result unchanged, if we do not, in making 

 this increase, ignore some known fact and so throw away 

 knowledge which we already have and which is pertinent 

 to the case in hand. We may illustrate by considering 



