192 MACKENZIE — EQUATIONS OF HEAT PROPAGATION. [April 4, 



medium of temperature initially zero. We can get a close 

 approximation to this problem by putting the same quantity of 

 heat, ffi into a particle of volume Av which we put into the math- 

 ematical point, and assuming that the state of temperature produced 

 in the surrounding medium is the same as that due to the infinitely 

 hot point and is given accordingly by (14), This equation will 

 represent the real state the better the longer the time which has 

 elapsed, in accordance with the fact emphasized by Fourier that 

 the initial heating is of less and less importance as the time is pro- 

 longed. The closeness of the approximation for any given time 

 and distance will be brought out later. 



Let the quantity of heat supplied raise the volume Av to the 

 temperature V^; then Q = CDV.Av, or a r= F,Jv; and (14) 

 becomes 



r- 



KAv ^^ 

 V= '—J e (15) 



If the volume Av is in the form of a sphere of radius R, (15) 

 becomes 



F,/?3 



\kt 



V=^^^^-^ e , (16) 



6i/7r (i/)i 



and it is really for this form of the equation, with R taken as the 

 unit of length, that the curves referred to on Plates XXV and XXVI 

 were drawn. They are, as said, approximations only to the true 

 curves. The latter may be found by the aid of a Fourier's integral. 

 We know that the solution of (2) subject to the condition V=^f{r) 

 when / = is 



Vr = ^ [J (^ + 21/17 r)/(/' -f Wk^r) 



e dy — 



•i}/'kt 



C» 2—1 



J (-r + 2i/I/r)A-^+ V^r)^ \/rJ--(i^) 



2\^ kt 



Giving f{f) the value V^ from r = to r =^ R, and the value 

 from r =^ R io r =^ ^ y {11) takes the form 



