20. Note on the Secular Cooling of the Earth and a 
Problem in Conduction of Heat. 
N. Mati, B.A., Sc.D., F.R.S.E., Prof., Presidency 
College. Communicated by Sir AsvTosH MUKHOPADHYAYA. 
In a celebrated memoir, Thomson (Lord Kelvin) appiet, 
as is well known, Fourier’s theory of conduction of heat 
determination of the probable age of the earth. The main line 
of his argument may thus be stated. 
The solution of the pe (in one dimension)— 
8 : 
== =k a v being the temperature, and ¢ the time, 
_knt 
is oP BAL m COS NX (1) 
where v, =the initial temperature = = A, cos nx. (2). 
Now, since v, is an actual temperature, the series on the 
right-hand side of (2) is necessarily convergen nt. 
.. the series (1) for v is also convergent for all positive 
values 0 
For negative values of t, however, the series may or may 
not be convergent and the particular value of ¢ for which the 
series ceases to ng convergent gives a superior limit to the 
admissible value of 
The above pute. therefore, rein on the assumption 
that the present distribution is due to a certain initial distribu- 
Witho ut considering the problem in its general form, we 
may pre sufficient information, as was shown by Thomson 
a later paper (‘On the Secular Coalags of the Earth’’) by 
sonaideents it in the following manner 
Let the regions separated by a = plans (a = 0) in an in- 
finite solid be initially at temperatures, v, 0, 
oe 
6 v, — UV, ae 
ee OR (9) 
