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



-^[^-/'"*]-S 



Vr 



2\/ kt 



\kt 



e ,..(8) 



and 



E 4^^ 

 V= ^e , (9) 



;- 



where £ is an arbitrary constant. 



Further light can be thrown on this problem by arriving at 

 equation (8) by other methods. Remembering that equation (6) 

 gave Fr initially infinite at the centre and zero elsewhere, and 

 falling in value at the centre and gradually rising to a maximum 

 elsewhere, we see that by taking the space doublet of this Fr we 

 get Fr 2it the origin first infinite and then zero; that is, Fat the 

 origin is at first infinite and then gradually falls off, and is initially 

 zero elsewhere and rises with time. These are the conditions 

 required. Hence the solution is 



d r A i^i-] Er ^kt 



^' ~ ^r \_VJt ' J 



{kty 



Or we can look at it in this way : We saw that Fr in (6) had 

 exactly the set of values we want F to have in the problem pro- 

 posed, and the form of the right-hand member of (6), containing 



r- 

 ~ \ki 



as it does r in the factor e only, suggests at once that we can 



get the desired value of Fby a simple differentiation with regard 

 to r. This is what we have just done with a good physical reason 

 for the operation. 



Or another method. We saw that equation (G) for the case of 

 flow in one direction only was that of an infinitely hot plane cool- 

 ing in an infinite medium initially zero in temperature, and to get 

 the solution for the similar problem in three dimensions we have 

 but to multiply that solution by two similar ones with y and z 

 substituted for x. This gives 



'^kt Akt \kt ^ Akt 



e e ^ --—5 e (11) 



{_ktf {.ktY- 



