100 PRINCIPAL FORBES ON AN EXPERIMENTAL INQUIRY INTO 



loss of heat follows Newton's law, or that the loss of heat in unit of time varies 

 simply as the excess of temperature ; (2.) That the same law holds for the internal 

 communication of heat, or that the quantity of heat conducted is proportional 

 simply to the difference of temperature of two adjacent elementary portions of 

 a bar. 



103. From the first assumption it follows, of course, that the temperature of 

 a cooling body of small dimensions varies in a decreasing geometrical progression 

 with the time. The dynamical Curve of Cooling on this assumption is a 

 logarithmic curve, t and v being the variables. 



104. From the second assumption, taken along with the first, we learn from a 

 well-known analysis, that what we have called the Curve of Statical Temperature 

 is also a logarithmic, x and v being the variables. 



105. Now the Statical Curve of Cooling (-r, in terms of x) must, on these as- 

 sumptions, be also logarithmic ; for its ordinates — the velocities of cooling — are 

 everywhere proportional to the temperature. Hence also the subtangent to 

 these two last curves* is the same. Let it be M. Then by a property of the 

 logarithmic curve (M being the modulus) the area of the curve bounded by an 

 ordinate y, and carried to infinity, is My. Also the flux of heat corresponding 

 to the position of the ordinate y is (Art. 99)= My, y being, as we have seen, 



= — y. But, by Art. 102, — -r is everywhere assumed (for the present) to be 



dv 



proportional to v, or — j =pv. Also since the dynamical curve of cooling is a 

 logarithmic (103), let its modulus be m. Then, by the property of the curve, 

 ~ dt = m' Hence, comparing the last two equations p = — . And, 

 F = Flux of heat = My = -M^ = M- 



^ at m 



and (by Art. 101). 



Conductivity = -j- = -r 



J dv dv 



dx dx 



But the curve of statical temperature being also assumed to be logarithmic (104); 

 and consequen 

 we finally get 



and consequently - -^ = =^; 



Mv M 2 

 Conductivity = = — 



J m v m 



' M 



106. A first approximation to the conductivity of the bar may therefore be 

 found by dividing the square of the modulus or subtangent of the statical curve of 



* Namely, the Curve of Statical Temperature and the Statical Curve of Cooling, being the two 

 curves shown in the wood-cut of last page. 





