THE LAWS OF CONDUCTION OF HEAT IN BARS. 101 



Temperature (assumed to be logarithmic) by the modulus of the Dynamical Curve 

 of Cooling. 



107. Thus, to illustrate this by a numerical example, were we to attempt to 

 reduce the statical curves of temperature of Table II. to logarithmics after the 

 manner of Biot, we should probably find the following approximate values of the 



subtangent M : — 











Case I. 



Case II. 



Case III 



M 



0-9 foot 



07 foot 



0-8 foot 



And from Table V. of the Dynamical Curves of Cooling, the subtangents might 

 be nearly 



m 60 min. 40 min. 50 min. 



whence 



— 0135 0122 -0128 



m 



which, it is seen, give nearly approaching values of the conductivity. 



§ VI. Final Determinations of the Conductivity of Iron at various Temperatures. 



108. The results given in the last section are in the' highest degree rude, and 

 are introduced merely to illustrate the general form of the method. The curves of 

 Temperature and Cooling are neither of them sensibly logarithmic, and therefore 

 we have found the necessity of dividing them into small portions, and taking their 

 elements from point to point. Therefore, in continuation of what has been said 

 in Art. 99, we must proceed to the quadrature of the Statical Curve of Cooling 

 whose elements are given in Table VIII. This is a curve which though not 

 logarithmic, may, like the other curves we have already discussed, be treated as if 

 it had been, when divided into numerous elements bounded by parallel ordinates. 

 Every one of these segments may have its area estimated by the simple formula 

 proper to a logarithmic curve,* and for the infinite branch a similar formula must 

 be adopted. 



109. The following Tables contain the determination of the total Flux of Heat 

 (F) across any section of the bar by the summation of the areas of the statical curve 

 of cooling, commencing from the colder end of the bar, where this curve is (like 

 the primary curve of temperatures) apparently asymptotic. In these Tables 

 (corresponding to the three experimental Cases discussed in this Memoir, the chief 

 uncertainty attaches to the two extremities of the curve. There are difficulties in- 

 herent in the precise determination of very small excesses of temperature of a bar, 

 whether in a statical or a cooling condition, above the surrounding air, itself not 

 absolutely constant in temperature. These difficulties have been previously referred 

 to. Moreover, when we have to take the ratio of two quantities, both to be experi- 



* Namely, area between ordinates y and ?/' = M (y'—y) where M the subtangent equals 



0-4343 &c. (x -so') 

 log y' -logy 

 VOL. XXIV. PART I. 2 E 



