1902.1 MACKENZIE — EQUATIONS OF HEAT PROPAGATION. ISD 



abscissa (time) the ordinates of all possible curves such as B^^ ^% 

 etc., below any given one, after reducing them as described. For 

 / = and r = we would get (co -|- -(- + etc.) dr, which as 

 ^r diminishes indefinitely gives us some finite value; for other 

 values of r we would get (0 -f- -f- -f etc.) dr, which is zero. 

 From the way the curves tend to become parallel it is suggested, 

 and by trial we find, that for r = and any finite value of the 

 time not zero the sum of all the ordinates would be constant. We 

 have then the promise of another simple solution, and can foretell 

 its type somewhat, of the form 



Vr 



CA^e ^''dr=^B C e d?, (7) 



2}/ kt 



where B is an arbitrary constant. On studying this equation we 

 find that Vr at the origin has initially the value B, and maintains 

 that value ; at all other points it is initially zero and rises asym- 

 ptotically with time toward the value B. F itself would be' always 

 infinite at the origin and initially zero elsewhere. For the case 

 of linear flow equation (7) represents an infinite plane kept at 

 temperature B in an infinite medium initially zero in temperature. 

 We can get the solution for an infinitely hot point put into an 

 infinite medium initially zero and left to cool as follows : At time 

 zero apply to the medium the state of temperatures represented by 

 (7) with every temperature increased by multiplying it by the large 



quantity—-; after time J/ apply also the state of temperatures 



represented by (7) with sign changed and increased numerically as 

 before ] finally make At tend to zero. We have seen above that 

 this is equivalent to performing the mathematical operation of 

 differentiation of (7) with regard to /, that is, taking the time 

 doublet of Vr. The reason that this solution is the one required is 

 that the superposition of the two heatings gives Vr a large value 

 at the origin at first and everywhere else a zero value, and then 

 instantaneously makes Vr zero at the origin ; that is, at the origin 

 Fis initially infinite in temperature and then falls off indefinitely, 

 while all other points begin at zero and rise gradually. These were 

 the conditions we wanted. Hence we have the solution 



