1902.] MACKEXZIE— EQUATIONS OF HEAT PROPAGATION. 187 



a solution of (1) represented by the mathematical operation V--' 



where Fr is a solution of (2) and F itself a solution of (1); we 

 have but to add to the doublet of this V as defined above a heating 

 at each point ;-, which is F divided by the value of r at the 

 point. 



The meaning of ( F dr, where F is a solution of the differential 

 equation, is now plain. It simply means finding a new function of 

 r and /, F^, whose doublet is the solution F. That is, -^ = F, and 



or ' 



/^i = (F dr. This is subject to the same limitations as before, 



that the differential equation for Fmust have its coefficients inde- 

 pendent of r, in order that F^ may be a solution of the equation. 



Similarly for equation (2); we have a solution, Fr, to find the 

 meaning of the new solution, FV, which we get on performing 



the integration J v r dr. Since -^^ — ' = Fr, or — — - =z F, we 



are but finding the distribution of temperature, F^, whose doublet 



added to the heating — gives the distribution of temperature, 



F, which we started with. 



We thus see that (2) has the great advantage over (1) that when 



we find a solution of the former we can differentiate and integrate 



it with regard to r for new solutions, but we cannot do so with the 



latter. 



dV r 



The meaning of -j- and of J Fdt as solutions of (1) are of the 



same general nature as the similar expressions with r, and are quite 



evident ; we now superpose one heating, — /(^/) on another, 



— - /ir,/"), after a small interval of time J/, which we make 

 smaller and smaller indefinitely. We might call this a //>;/<? doublet 

 and the former a space doublet. Both ^ and J Fdt are solutions 



of (1) because the coefficients do not contain t. The same remarks 

 apply to (2) as regards Fr, with the explanations of the former 

 paragraph added. Here equation (2) possesses no advantage 

 over (1). 



The meaning of a Fourier's integral may now be given. A 

 solution of (3) for the flow of heat in one dimension is evidently 



