1902.] MACKENZIE — EQUATIONS OF HEAT PROPAGATION. 191 

 The rate of cooling is given by the equation 



Each point of the mass not the centre begins by being zero in 



temperature, then rises to a maximum after a time / ^= -^ , and 



after this falls off indefinitely toward zero. The forms of the 

 curves given by (9) are exhibited on Plates XXV and XXVI. With 

 values of r as abscissae curves I^ to IV' are for values of the time 



jTw, -Q^, -j^, and —J respectively ; with values of 4/'/ as abscissae 

 curves I' to 5^ are for values of the distance 0, i, 1, |- and 2 

 respectively. 



The meaning of the constant £ is determined by finding the 

 amount of heat supplied initially to the hot point. We have 



(2 = r r r CC> Vdx dy dz = ^^^^^ C e ^^\-\ir=-^ CDfJ. . (12) 



If we take as our unit of heat that required to raise the niass in a 

 unit of volume of the substance 1°, the total quantity of heat, (t, 

 in these units is 



ff = 8^-' (13) 



We could also get the total heat by taking the integral 



C— K ^ i-r'dt. We get from (12) and (13) our equation (11) 







in the form 



Q 4^< ^ ^kt 



y — . ^ ^ ^ £ C14) 



^CDi^-Kktf. 8(-'^0^ 



(See Kelvin's Papers, Vol. II, p. 44.) 



We cannot build up by summation the solution for the case of a 

 body of finite dimensions from the above solution for a mathemati- 

 cal point. We wish to pass to a case which has a physical signifi- 

 cance, namely, a finitely hot particle left to cool in an infinite 



