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



In any heat conduction problem we have ordinarily three sets of 

 equations, the general differential equation, the initial conditions, 

 and the surface conditions. For the general purposes of this 

 paper by taking the medium infinite we can get rid of the surface 

 conditions without limiting the generality of the methods. Suppose 

 we wish to study the case of a body of any shape or size maintained 

 at any temperature in an infinite homogeneous medium of the same 

 material as the body itself but initially at a uniform low tempera- 

 ture (which for convenience we take as the zero of temperature), 

 or of the same body at a given initial temperature put into the 

 medium and left to cool, we could find their solutions by an 

 ordinary summation if we knew those for the corresponding prob- 

 lems in the case of an infinitesimally small particle. We might 

 begin by assuming as Kelvin does {Math, and Fhys. Papers, Vol. 

 ii, p. 44), the solution for the case of a quantity of heat, Q, sud- 

 denly generated at a point r = at time / = ; but it will be 

 better to see if it can be derived. 



We have here to deal with the case of a symmetrical distribution 

 of temperature about a point. The form of the general differential 

 equation for this case is 



l^^Z=z^ 5F 52^ (I) 



k dl r dr dr^ 



where k = ^r^, J^ being the specific conductivity, C the specific 



heat, and D the density of the medium. This equation can be 

 put in the more symmetrical form 



1 ^AZ^ — ^'(^'^'') (2) 



This is of exactly the same form as that for the case of the 

 *' linear flow of heat " of Fourier, that is, of flow in one dimen- 

 sion only, namely, 



J_ 5 F _ d'^V (3) 



The distribution of Vr with reference to r for the case of sym- 

 metry about a point is the same as the distribution of V with 

 reference to x for the case of symmetry about an infinite plane 

 perpendicular to the axis of x. This fact will be of assistance in 

 obtaining and translating results. The ordinary way of treating 



