FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 151 



Consistently with our numerical discussion we have not tabulated 

 the thermal properties nor the thickness of the specimen itself, since 

 the effect of these is vanishingly small. 



We have now two cases to consider; first the steady temperature 

 rise. The period of the impressed heat input does not enter, and we 

 have to find all the dimensionless products of the first six quantities of 

 the list above. Since there are four fundamental kinds of quantity 

 (instead of unit quantity of heat H, we might have expressed heat in 

 mechanical units, thus replacing // by M, with no change in the final 

 result), and hence two dimensionless products. Inspection shows 

 these products to be kr/Q and k/gch-. Hence the relation which we 

 want may be expressed as 



urn 



where / is some unknown arbitrary function. This relation can be 

 tested by experiment, and so some idea obtained of the correctness of 

 the assumptions underlying the discussion. For instance, at constant 

 rate of flow of cooling water, the above equation shows that the steady 

 temperature rise should be proportional to the rate of heat input, or 

 to /-. We can obtain an additional check for low rates of flow. For 

 low rates, but not too low, it seems natural to assume that an impor- 

 tant part of the rise of temperature is inversely proportional to the rate 

 of flow, or inversely as g. This means that in / there is an important 

 term which is the reciprocal of its argument, and we obtain as a partial 

 expression 



• T = Const Q/gclr'. 



Some experimental information may be obtained here by varying 

 the breadth of the sample at approximately equal rates of flow. That 

 the average rise of temperature should be less for the greater breadth 

 seems somewhat paradoxical, and affords a more drastic test than the 

 proportionality of temperature rise to the rate of heat input. 



Now let us consider the alternating fluctuations of temperature. 

 To distinguish from the steady case, and consistently with the previous 

 notation, we denote the amplitude of the alternating heat input by Qi, 

 and the amplitude of the alternating temperature change by n. We 

 now have seven quantities, and hence three dimensionless products. 

 The additional product, beside the two already obtained, is gr/co. The 

 relation between the variables now takes the form 



