Passing Isothermally from Solid to Liquid. 9 



Method of computation. 



9. General case of the environment. — A short resume of the 

 changes of condition involved is here necessary, since for 

 reasons specified in §2, I found it necessary to depart somewhat 

 from the method of Weber. 



In his masterly discussion of the flow of heat in the plates 

 Weber shows that the two copper discs are isothermal regions 

 identical to about 1 : 1000 with the upper and lower isothermal 

 surfaces, respectively, of the enclosed non-metallic liquid, no 

 matter what it be. Hence in addition to Fourier's well known 

 equation of heat conduction, the following surface equations 

 obtain for the upper plate : Given a set of cylindrical coordi- 

 nates whose origin is in the upper surface of the lower plate, 

 and whose axis coincides with the axis of the plates ; let dis- 

 tance above the surface of reference, radius and azimuth of 

 any point, whose temperature excess is u at the time t, be rep- 

 resented by x, r and <p, respectively. Let r = i? be the radius, 

 J x the thickness, F x the exposed surface (top and sides), i^the 

 lower surface of the upper plate, M x its mass, c 1 its specific 

 heat, and h x its external conductivity. Let J be the thickness 

 of the layer of the charge, k its absolute heat conductivity, h 

 its external conductivity, c its specific heat. Finally let U be 

 the uniform temperature excess of the system at the time zero. 

 Then the conditions in question are 



1. x = 0, u = for all values of t. 



2. x = A, a independent of r for all values of t. 



3. x = A, - M A (^) A =k j(aj) 4 + M> A 



4. r = Ii, k(du/dr) -i-hu^ =0 



5. t = 0,u=.lf, for all the values of x and r. 



Weber expands u in a series of mixed Bessel functions of the 

 type 



—TcqH/pc . -Jc(p - + m*)t/pc . 



u = Ae sin qx + JJe &mpxJ Q (mr) 



and proves that by suitable spacing the copper discs, the value 

 of u is almost wholly contained in the first term even after 

 a few seconds. To determine the constant q, the insertion of 

 the condition (3) is available : i. e., any of the infinite roots of 

 the transcendental resulting, leads to a singular solution of the 

 equation of heat conduction, and thus the complete primitive 

 is given as the sum of all of these. But inasmuch as the 

 squares of the succession of values q, q', . . . . increase rapidly, 

 the singular solution and the complete primitive again soon 

 coincide in the lapse of time. Thus the temperature excess of 



