THE LAWS OF CONDUCTION OF HEAT IN BARS. 87 



71. I cannot too distinctly repeat that the formulae adopted in the preceding 

 Table are only to be regarded as a means of more conveniently grouping the 

 observations. The most important use to be made of these formulae, however, 

 yet remains to be mentioned. It will be seen by reference to Arts. 6, 28, &c, of the 

 former part of this paper, or to § IV. of the present paper, that it is not the ordi- 

 nates themselves of the statical curve of cooling which are to be used in obtaining 



the conductivity of the bar, but the values of the differential coefficient - ~ for 



each part of the bar. In other words, we must be able to draw a tangent to the 

 curve of statical temperature at any point of the curve. This may be roughly 

 done mechanically, or it may be done by dividing the curve into short elementary 

 portions, and treating each portion as if it were part of a logarithmic curve (see 

 below, Art. 82, on the Analogous Treatment of the Dynamical Curve) ; or, 

 finally, it may be obtained from the equations above given. The two last methods 

 have been used in the reductions, and especially the last of all, which is the only 

 satisfactory one for the higher parts of the statical curve. The general form of 

 the empirical equation being, 



Tab. log v = A ■ 



1 + CX 



when reduced to Napierian logarithms, gives 



7)7* 



0-4343 hyp. log v= A 



1 + ex 



and ?= -2-3036, 



dx (1 + ex) 2 



whence the numbers which will be given in § IV. of the present paper are com- 

 puted, the values of b and c being taken from the formulas of Art. 69. 



§ II. Experiments on Cooling. 



72. The Apparatus. — It will be seen, by reference to the former part of this 

 paper (Arts. 5, 24), that, in order to interpret the indications of the permanent 

 temperature of a bar, and to deduce its conductivity, we must have an inde- 

 pendent set of observations on the cooling of a similar bar, or a portion of a 

 similar bar. For this purpose, the apparatus shown in fig. 2 of Plate I. was 

 employed. The same short bar, L M, which has been already referred to (Art. 

 51), as being used in the statical experiment for determining the temperature 

 the bar would have had independently of the heat applied at one end, was sup- 

 ported on the props N, 0, after being duly heated. It is now to be used to ascer- 

 tain the rate of loss of heat from a bar having the Section and Surface proper 

 to each of the three Cases of Art. 48, in terms of the scale of the thermometer P, 

 inserted at or near its middle point. 



73. I have so fully described, in Art. 24, the manner of performing the 



