284 LAMBERT— ON THE SOLUTION OF [April 20. 



If the solution of the three-dimensional Fourier's equation 

 dF „( d-V d'V d^F 



/ a-F d'F d'F\ 



dt 



is a function of ;• and / only, so that 



F = f{r, t) , where r = (.r- -f y- -\- c- ) *, 



the transformation of the given equation from rectangular to polar 

 coordinates shows that the solution is 



r 

 where u is a solution of the Fourier's equation 



du d'^ti 



It follows that solutions of the three-dimensional equation of the 

 form F = f(r, t) are obtained by replacing .r by r in any solution 

 of the one-dimensional equation 



aF _ dht 



and dividing the result by r. 



In this manner are obtained the solutions 



V(i) F=4. 



(2) F=^— -- 



(3) F = — e"' sin (;/;- -|- 2irKt). 



It is interesting to compare the solutions of Fourier's partial dif- 

 ferential equation obtained in this paper with the solutions tabulated 

 by Sir William Thomson in the mathematcal appendix of the article 

 on "Heat" in the "Encyclopaedia Britannica," ninth edition. 



Sir William Thomson obtains his results bv summation, that is 



