1902.] MACKENZIE — EQUATIONS OF HEAT PROPAGATION. 183 



any problem of spherical symmetry is to get the simplest kind of a 

 solution of (1) or (2) and build up from that solution to the 

 required one. There is of course an infinite number of solutions 

 of these equations and a great many simple ones, but we can at 



once find one by trying Vr =^ e . This gives /5 = ko\ and 



ar kd-t —ka-t 



hence Vr =^ e e . Changing a to ia we get Vr = e 

 (cos ar + / sin ar), and so a solution is 



Vr=^e cos ar, (4) 



where a is any constant. This equation represents a periodic dis- 

 tribution of Vr along a radius vector dying out with the time ; lor 

 the case of the infinite plane this would be actually the curve of 

 distribution of temperature along x. It is seen that the values of 

 Fin (4) possess maxima and minima; the temperatures are zero at 



distances given by ^ = (2/z + 1) -^ at all times. There is a hot 



central sphere of radius ^-, surrounded by alternate hot and cold 

 shells of common thickness — , the maximum numerical tempera- 

 ture in each falling as we go away from the centre. Calling the 

 thickness of the shells d, we have a = ^ j so that the constant a is 



inversely proportional to the thickness of the shells and deter- 

 mines it. The central point begins by being, and remains, 

 infinitely hot ; the hot and cold layers conduct heat to each other 

 and gradually die down in temperature. At a great distance from 

 the origin we should have practically the case of a medium made 

 up of alternate hot and cold infinite plates of the same numerical 

 temperature and the same thickness left to cool ; and such a prob- 

 lem could be treated from a consideration of (4). 



This case is far from the problem we started out to discuss. We 

 can, however, get new solutions from the simple one above, and 

 the common method is now to say that the following is a solution 

 of (2), 



e cos ar da, (5) 







and then translate this equation as we have just translated (4) ; but 



