574 
MLi. R. W. STEWART ON THE ABSOLUTE THERMAL 
of the temperature curve for the copper bar, and fig. II gives the rq^per part of the 
coolingf curve for the iron bar. 
The experimental results obtained and the details of their reduction are set out in 
the Tables given below, which, taken in connection with the following general expla¬ 
nation of the notation adopted, indicate the successive steps of the reductions. 
Let h denote the absolute thermal conductivity at any normal section of the bar at 
a distance x from the heated end, and where ddjdx is the gradient of temperature. 
Then, if a denote the area of cross-section of the bar, ha [ddjdx) denotes the heat 
which crosses the section considered in unit time. This heat is lost by radiation 
from the surface of the bar beyond this section, hence, if I denote the length of a 
short element in this part of the bar, 9 its mean temperature, and ddjdt its rate of 
cooling at this temperature, then the heat lost per unit of time from the surface 
of this element is lasd [dOjdt), where denotes the specific heat, and d the density of 
the material of the bar at the temperature 6. From this it follows that, if all the 
units involved be consistent, we have 
ha {ddjdx) = S lasd [ddjdt) .(l). 
Introducing corrections for temperature this relation becomes 
/v (1 + 2oid) [ddjdx) = [ddjdt) 
where /y, d^ are the values of I, a, d at 0° C., d the temperature of the cross 
section of the bar, and a the coefircient of linear expansion of the metal. The value 
of h for any cross-section can evidently be found directly from (2) by calculating the 
corresponding value of %IqS [ddjdt), where 5 has its proper value for each element 
of the barthe reduction can however be more conveniently made by taking 
IIqS [ddjdt) as erpial to s%Iq [ddjdt), where s is the specific heat at a mean tem¬ 
perature d, which is given by 
This gives us 
or 
d = [S/o [ddjdt) d'\jltk [ddldt)\ 
^■(1 + ‘lad) [ddjdx) = dQs'Sl^ [ddjdt), 
7. _ 'b 2; h (ddldt) _ J /q (ddjdt) 
l+2u0 {ddjdx) (ddjdx) ’ 
where M denotes the factor involving s‘. 
In the reductions given below it will be seen that the absolute conductivity, h. 
* [An eiTor ooiniccted with this point, in the results first communicated to the Royal Society, w 
pointed out to mo and has now been corrected.—R. tVh S., 13.7.93.] 
