702 Mr. E. Buckingham : JS J otes on 



The II theorem accordingly tells that equation (9) is 

 reducible to the form 



f(M =0 > •••••• ^ 



or, i£ N is a root of this equation, 



fe = ^ (11) 



N being a pure number. 



Stated in words, the meaning of equation (11) is that if 

 the density of a fluid is completely determined by its 

 pressure, temperature, and specific heat at constant volume, 

 and if, further, the specific heat is constant, the fluid is an 

 ideal gas. Or, to put it the other way about : the specific 

 heat of an ideal gas is constant. 



As another possibility, let it be assumed that the new 

 thermal quantity required for completing equation (8) is 

 the thermal conductivity \, with the dimensions \inlt~ z 6~ l ~\. 

 If we put this into equation (8) and try to apply the 

 II theorem, we find at once that there is no solution and 

 that the equation is still incomplete. Let us see what 

 results from introducing, also, the viscosity p, with the 

 dimensions [ml~ l t~ 1 '] and writing 



F(p,p,0,\») = O (12) 



This equation may be complete, for the II theorem gives 

 as a solution 



f(-m=^ en. 



or, if N is a root of this equation, 



Z n = ^~ (14) 



p0 p 



If, therefore, there is a complete relation of the form (12), 

 the value of p/pO is proportional to that of ~\//j,. For ideal 

 gases, the ratio of thermal conductivity to viscosity is 

 constant ; or if this ratio is constant, the fluid is an ideal 



In either of these cases, all we can say with certainty is 

 that equation (8) is incomplete. Whether (9) or (12) is 

 complete we do not know, a priori, but merely that either 

 of them may be. If either of them is, then we have the 

 result expressed by (11) or (14), which may be compared 

 with observation. 



