THE LAWS OF CONDUCTION OF HEAT IN BARS. 83 



65. It will be observed that in every instance, with the single exception of the 

 final number of the Table under Case III., the decrement of temperature becomes 

 materially slower as we recede from the heated end of the bar. The exception 

 is of little weight, as it depends on the residual temperature of the one-inch bar, 

 six feet from the crucible, at which point the warmth was barely perceptible. 

 The statical temperatures of the bar therefore increase more rapidly than in a 

 geometrical progression. I have found that there is a sufficient analogy between 

 the curve of statical temperature which we are here investigating, and that of 

 the tension of steam at different temperatures, to afford some assistance in the 

 selection of empirical formula? in the present instance ; being in each case a 

 modified geometrical progression, though here the progression of ordinates is 

 more rapid than a simple continued proportion, while in the tension of steam 

 it is less rapid. 



66. The most complete discussion of this class of formulae, and the methods 

 calculating from them, is to be found in M. Regnault's Relation des Expe- 

 riences sur la Vapear. The available formulae are reducible to three ; Young's,* 

 Roche's, f and Biot's4 The two former contain three constants, the latter five. 

 The last has been found the most satisfactory for the empirical representation 

 of the elasticity of steam; and it is the only one of the three which can be 

 regarded as applicable throughout the entire limits of experiment. The same is, 

 I believe, true for the Conduction-curves with which we are now occupied. With 

 five constants, five points of the curve may be accurately represented, and the 

 intermediate deviations are of course inconsiderable. As none of the formulae 

 (except the simple logarithmic which is found to be inapplicable) have any foun- 

 dation in principle , the whole matter is purely one of convenience. For simplicity's 

 sake, I used only the formulae of Young and Roche, but I now think that 

 the greater labour involved in the application of Biot's formula would have been 

 repaid by the directness and certainty of the results. M. Regnault has given 

 rules to facilitate its numerical calculation. 



67. I have found the formula, — 



lo s"= A -rrW • • • • ■».(!■) 



(where v is the excess of temperature above the air at a point of the bar, whose 

 abscissa, in feet, is a, and A, b, and c are constants), to represent tolerably well 

 the temperature curve of Case I., as represented by the numbers in Table II. , 



* p — A (1 +at)'\ where p is the elasticity, and t the temperature, 

 t log p = log a + ^-~ t . 

 X log p = a + ba' + cj8'. 



