of Edinburgh, Session 1880-81. 
129 
The equation for the consequent distribution of temperature is 
If we assume 
where a and /3 are small positive constants ; 
and put 
V = U + (O , 
where w depends upon first powers of a and /3 only, higher powers 
being neglected ; the equation splits into two as follows : — 
which satisfies (1), and shows the ultimate effect of a persistent 
simple harmonic application of heat to one side of the slab, whose 
temperature is taken as our temporary zero ; the other side being 
kept at the temperature - Bs, where s is the thickness of the slab. 
Here s must be supposed so large that Ce" ws is insensible; else 
the value of u would be so complicated that (2) would become un- 
manageable. 
Substituting the above value of u in (2), and integrating, we 
obtain the value of w. It consists of three parts. 
We have, first, terms containing x only : — 
These terms show how the mean temperature is altered throughout. 
Next, we have the single term 
du d 2 u 
(!)• 
dt K dx 2 
For our present purpose it is sufficient to take 
u— - Bx + Ce~ mx cos (2 Km 2 t - mx ) 
$ B 2 ^ + ^CV 2 “ 
This is a small wave of half period, which we need not farther con- 
sider. 
