364 MR T. C. BAILLIE ON THE 



§ 3. Reduction of the Readings. — The equation for the conduction of heat in a 

 bar, each part of which is at a steady temperature, is : — 



KA^ = Ep0, 



where K is the thermal conductivity, A the cross-section, E the emissivity, and p the 



perimeter of a part of the bar, at temperature 6 above the surrounding air, and at a 



distance x from some fixed point in the axis of the bar. Since K, A, E, and p are 



d 2 9 

 either constants or functions of 6 only, it follows that -j— 2 is a function of 6 only, and 



therefore the value of d 2 6/dx 2 for any given value of 6 should be the same, no matter 

 which of the sets of readings it is derived from. This affords a means of testing the 

 concordance of the various sets of readings. The determination of d 2 9jdx 2 directly— 

 by drawing a curve representing 9 as a function of x, taking the tangents at various 

 points, and thus getting another curve showing dd/dx as a function of x ; and from this, 

 by a similar process, another showing d 2 6/dx 2 as a function of x, and using the first and 

 last curves to get d 2 0/dx 2 as a function of 9 — does not give d 2 0/dx 2 with sufficient 

 accuracy. A common proceeding is to find suitable values of the constants in some 

 empirical equation representing 9 as a function of x, and to differentiate the equation to 

 obtain dO/dx. The following method of reducing the readings obtained in the statical 

 experiment was adopted after trying others. Curves were made from the sets of read- 

 ings on the first four thermometers only, in which log. 9 was shown as a function of x. 

 The gradient of these curves increased, but not very rapidly, with log. 9, and therefore 

 d(\og. 9)/dx increased as increased. The curves were drawn by a lath, to the ends of 

 which couples were applied so as to give it the slight curvature necessary to make the 

 curve produced by its means pass in close proximity to each of the four points given 

 by the corrected readings of the thermometers. It was noticed that the value of 

 d(\og. 6)/dx was practically the same, for the same value of 0, for all curves. A new 

 curve was then constructed, in which d(log. 0)/dx was shown as a function of 9. The 

 different points found on this curve lay very approximately in a straight line — that is 



to say, -jq( ^ f ' — -j was practically constant. The equation to the statical curves 



1 a 



must then be of the form j-log.A— y = a: + B, where b and c have the same value 



be ° 9 + b 



for each curve. The simplicity of this method of finding the average values of d l 9jdx* 



for all sets of readings was what led to its adoption. In any case, d(\og. 6)/dx does not 



vary so rapidly as ddjdx, and it is therefore easier to get dO/dx with accuracy, when 



using graphical methods, by multiplying d(\og. 6)jdx by 0, than it is to get dQjdx i 



directly. The values found for the constants in the above equation were c - '0000505, 



and 6 = 670. The value of d 2 6/dx 2 is c 2 (20 + b)(e + b)6. The following table contains 



the values of d 2 6/dx 2 calculated, not from the expression just given, but from the 





