the Method' of Dimensions. 701 



Hence one dimensionless product II may be formed, and 

 the equation must be reducible to the form 



-A. 



pX 3 

 M 



(7) 



The necessary conclusion is that if the index of refraction 

 depends only on wave-length, density, and molecular mass, 

 it does not depend on them separately but is fixed by the 

 value of (pA, 3 /M'j, i. e. by the number of molecules contained 

 in a cube of edge X. It must then be possible to determine 

 the effect of compression from the dispersion curve, and vice 

 versa ; and if this is not found to be true, experimentally, it 

 is certain that equation (6) is not complete. 



5. Second example of Incomplete Equations : the 

 Equation of State of Fluids. 



The density p of a fluid depends on the pressure p and 

 the absolute temperature 6 ; but the equation 



F( P ,p,d) = (8) 



is not complete. For it is evident upon inspection that there 

 is no dimensionless product containing 6 ; and since we 

 know from experience that all three of the quantities do 

 enter into a mutual relation, so that none of them can 

 be omitted, one or more additional quantities must be 

 introduced : there is no general equation of state for all 

 fluids involving these three quantities and nothing else 

 except dimensionless numbers. 



Since equation (8) is not complete, dimensional reasoning 

 cannot give any information as to its more precise nature. 

 To go farther, it is necessary to introduce at least one more 

 quantity ; and if there is to be only one, its dimensions 

 must be expressible in terms of density, pressure, and 

 temperature as fundamental units. Not knowing, a priori, 

 what the missing quantity or quantities may be, let us 

 assume that the specific heat at constant volume, (V, also 

 enters into the relation, and see what comes of assuming 

 that the equation 



¥(p,p,0,G v ) = O 19) 



is complete. 



On the m, /, t, 6 system, the dimensions of the four 

 quantities are [p\ = [?nZ" 3 ], [p] .= [mi" 1 *-*], [0] = [0], 

 and [C„l = [l 2 r 2 0- 1 ] ; and pOGp/p is a dimensionless 

 product involving all four quantities, so that equation (9) 

 may be complete, whereas equation (8) was certainly not. 



