700 Mr. E. Buckingham : Kotes on 



are of serious importance. Whether the assumption is 

 correct and the initial equation sufficiently complete can, 

 in general, be decided only by comparing the results of 

 theory with those of experiment. But in some few 

 instances an oversight in making out the list of quantities 

 is immediately evident upon inspection : cases of this sort 

 may be illustrated first. 



4. Incomplete Equations : test by inspection. 



Let it be supposed that when the initial equation tor 

 some problem has been written down, it is found that 

 the dimensions of one of the quantities involve a funda- 

 mental unit that does not appear in the dimensional 

 equations of any of the other quantities. It is evident 

 that there can be no dimensionless product of this quantity 

 with any of the others, and it follows either that the 

 equation is incomplete or that the quantity in question 

 does not, in fact, enter into the relation at all and might as 

 well have been omitted. 



To illustrate, let us consider how the index of refraction n 

 of a certain medium, for light of wave-length X, may depend 

 on the density p and temperature 6 of the medium ; and let 

 us Avrite, tentatively, 



F (n, \,p,8) = (4) 



We see at once that there is no dimensionless product of 6 

 with >v and p : hence either 6 ought not to appear, or some 

 other quantity involving temperature should be included in 

 order to make the equation complete. 



Assuming that temperature affects the index of refraction 

 only indirectly, through its influence on density, we may 

 omit and write, as the initial equation, 



F O, X, p) = ; (5) 



but this is not yet satisfactory. For there is no dimensionless 

 product of \ and p, and the equation requires, for com- 

 pleteness, the inclusion of at least one more quantity. 



To make a very simple hypothesis, let it be assumed that 

 the missing quantity is the mass M of a single molecule, so 

 that the initial equation is 



F (n, p, X, M) = (6) 



This contains, besides the dimensionless number n, three 

 dimensional quantities requiring two fundamental units. 



