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

 Hence the equation (3) may now be written 



g = 5+2{Prf^»_l) + D**r-}, . . (10) 



or, say, for shortness, 

 d 2 



dot 



= {R(a 9 -l) + S0 l +'\(m + 0). . . . (11) 



This equation has now to be integrated. Whether it can be 

 integrated completely as it stands I do not know ; but a first 

 integration is easy, and the result is 



17. In this expression the integration has been performed 

 between the limits and ; or one may say that the arbitrary 



constant has been chosen so as to make — and vanish together 



— which it is evident they do in an infinitely long rod, from 

 physical considerations. This is the object of using a long 

 rod. The experiment would be easier to carry out with a short 

 rod or a ring heated at one end and cooled at the other ; and the 

 differ ential equation (1) would apply equally well ; but in this 



case -j- would have a finite value when = 0, the right-hand 

 aoc 



side of (12) would not contain as a factor, and the whole 



calculation would become more complicated. 



Statical Curve of Temperature in a Vacuum. 



18. Equation (12) consists of two parts — the R part rela- 

 ting to radiation, the S part to convection : in a vacuum S is 

 zero. In order to do any more with the equation, I must take 

 the two parts separately. The radiation part is the simpler. 

 So let us suppose the rod on which the curve of temperature 

 is being observed has a blackened surface, and is in a perfectly 

 exhausted enclosure. The convection part I have only slightly 

 attacked at present, and have not succeeded in doing any thing 

 practical with it. I am unable to integrate even the radiation 

 part of (12) any further as it stands ; so at this stage we will 

 introduce our very approximate expansion (7') for a -— 1; and 





