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XXXVIII. Problems in the Conduction of Heat. 

 By Lord Rayleigh, O.M., F.R.S.* 



THE general equation for the conduction of heat in a 

 uniform medium may be written 



dv _d 2 v d 2 v d 2 v _ 2 



di~d^ 2 + oy + d^~ va ' m ' ' ' w 



v representing temperature. The coefficient (v) denoting 

 diffusibility is omitted for brevity on the right hand of (1). 

 It can always be restored by consideration of " dimensions." 

 Kelvin | has shown how to build up a variety of special 

 solutions, applicable to an infinite medium, on the basis of 

 Fourier's solution for a point source. A few examples are 

 quoted almost in Kelvin's words: — 



I. Instantaneous simple point-source ; a quantity Q of 

 heat suddenly generated at the point (0, 0, 0) at time £ = 0, 

 and left to diffuse through an infinite homogeneous solid. 



where r 2 = x 2 -\- y 2 -\- z z . [The thermal capacity is supposed 

 to be unity.] Verify that 



I \v dx dy dz — 4V I vr 2 dr= Q ; 



— CO 



and that i?=0 when t = ; unless also x = 0, y = Q, c = 0. 

 Every other solution is obtainable from this by summation. 



II. Constant simple point-source, rate q : 





(3) 



The formula within the brackets shows how this obvious 

 solution is derivable from (2). 



III. Continued point-source ; rate per unit of time at 

 time t, an arbitrary function, f(f) : — 



* Communicated by the Author. 



t Compendium of Fourier Mathematics, &c, Fnc. Brit. 1880 ; 

 Collected Papers, vol. ii. p. 44. 



