the Method of Dimensions. 709 



geometrical form of the bearing and the ratio of clearance 

 to diameter. 



Let us now, as an alternative procedure, suppose that 

 heat is measured by an independent arbitrary unit o£ its 

 own, which we may denote by h. Joule's equivalent J 

 must now evidently appear in the initial equation, which 

 becomes 



F (A, D, o-,>, \, J) = 0, . . . . (36) 



containing ?i = 6 quantitities. But the number of arbitrary 

 units is now k=5, so that there is only one II, as before. 

 In terms of the 5 fundamental units, [\] = \_hl- 1 f 1 0~ 1 '] 

 and [J] = [mlH~ 2 h~ r \, and the solution of (36) by the 

 II theorem is 



'G^H* <^ 



hence 



N^, (38) 



which becomes identical with (35) if J is a constant. 



In the first treatment of this problem, the known constancy 

 of J was utilized to determine a heat unit, so that only 

 4 fundamental units were needed. In the second, J was 

 treated as a dimensional quantity and its constancy was not 

 made use of for eliminating the unit of heat. Hence 5 

 instead of 4 arbitrary or fundamental units were required. 

 This is quite analogous to the result with the oscillating 

 spheroid. In both problems we get, according to how 

 we start, either a restricted result, applicable only to our 

 universe, or a more general one which would be applicable 

 if: the "universal constant " were not constant but variable. 



10. General remarks on Universal Constants. 



The foregoing problems illustrate the manner in which 

 any universal constant is to be treated when the fact that it 

 is constant is one of the essential characteristics of the 

 phenomenon under examination. This fact must be taken 

 into account, but there are two ways of doing so. We 

 may start by treating the constant as a dimensionless 

 number and so eliminate one of the independent funda- 

 mental units that would otherwise be required ; or we may 

 put the constant into our initial equation like any other 

 quantity that is known to be involved, carry it all through 

 the algebraic work, and finally cross it out at the end, as 

 a constant. Algebraically, the first method is a little the 



