698 Mr. E. Buckingham : Rotes 



on 



2. Nature of the results obtainable from Dimensional 

 Reasoning. 



Dimensional equations are conventional short-hand de- 

 scriptions of general relations which subsist among physical 

 quantities of different kinds, and the distinguishing pecu- 

 liarity of dimensional reasoning is that it uses the known 

 facts of Physics only in this general form. Instead of 

 making detailed quantitative assumptions about the pheno- 

 menon under consideration, we make only the qualitative 

 assumption that it may be adequately described in terms of 

 a certain set of quantities — the Q's of equation (1) — , the 

 quantitative dimensional relations among these quantities 

 being matters of certainty. 



It follows from the nature of this process that the results 

 obtained by it will always contain undetermined numerical 

 constants ; for the dimensional equations employed also 

 contain undetermined numerical constants, although it is 

 customary to set these constants equal to unity and suppress 

 them. But on the other hand, if the original qualitative 

 assumption is correct, the general form of the result obtained 

 is an algebraic necessity and is certainly correct. 



As regards accuracy, it may be remarked that the results 

 of dimensional reasoning are subject to the same limitations 

 as those of any other theory. Theory always operates on an 

 ideally simplified picture of reality because real phenomena 

 are unmanageably complicated. The results obtained are 

 not exactly true for any real phenomenon, though thev 

 may be for an ideal one ; and the approximation with 

 which a theoretical equation, however obtained, represents 

 the actual facts, always depends on the approximation in 

 essentials between the ideal picture and its real prototype. 



3. General plan of attack on a specific problem. 



The purpose of dimensional reasoning is to find out 

 how some quantity which is involved in the phenomenon 

 under consideration is related to certain others ; or to 

 find the relation connecting two or more quantities which 

 vary, or may vary, simultaneously during the course of 

 the phenomenon. Since we know that we must have a 

 complete equation to start with, Ave begin by thinking the- 

 matter over, to see whether the quantities we have in mind 

 are the only ones involved. Usually it is evident that 

 they are not ; so we next make a list of all the quantities 

 we can think of which might under any circumstances be* 

 important. Upon considering this list, which is often a 



