200 Dr. 0. J. Lodge on the Variation of the Thermal 



cpqdx; but when the permanent state is reached, the heat 

 gained and the heat lost become equal, and their equality is 

 the fundamental differential equation for the permanent state 

 of a rod, viz. 



kqd -j- = Updx, 



or 



dx 2 kq ' 



The four quantities which enter into the right-hand side of this 

 equation are all variables, and may be expressed as functions 

 of 0. It has, however, been always assumed, in the approxi- 

 mate theory hitherto used, that H is the only variable, and 

 that it is simply proportional to the excess of temperature, and 

 can be written 



-R=h6, 



where h is a constant. (This is called Newton's law.) What 

 we now want to do, however, is to take into account the vari- 

 ability of all these constants as far as present experimental 

 results will enable us to do so, and then to integrate the above 

 equation to as great a degree of accuracy as is easily possible. 

 3. Now, if an isolated body of volume V, surface 8, density 

 D, and specific heat loses from each unit of surface a quan- 

 tity of heat H per second, then its rate of fall of temperature 

 is 



_dv ._ SH 

 dr° V V ~YJ)C ; 



' . dv 



writing v or 6 for the essentially positive quantity — — . Hence 



aT 



an element of the rod (§ 2), if isolated from its neighbours by 



two flat impervious films, will cool at the rate 



/b_ E-pda 

 cp dx 

 whence its rate of loss of heat per unit of surface is 



H=2^0. (2) 



Substituting this value of H in equation (1), it becomes 



S=f * o» 



which is precisely the form of the equation to the variable 



flow of heat through a slab*, though 6 has there a very diffe- 



* See Everett, Trans. Roy. Soc. Edinb. vol. xxii. 



