1902.] MACKENZIE — EQUATIONS OF HEAT PEOPAGATION. 185 



operation (5) therefore promises at least another simple solution 

 and one much nearer the desired one. Noting that 



^'■^—ko.'^t /»^ —koT-t 



J-r^—kol-t /»^ —koT-t 



e cos ar ^a = 2 I e COS ar da, and that 

 



+ »3 +00 



J—kaH /» —{kto:-—ira) 



e sin ar da = 0, we get \ e da — 



— 00 — cc 



r'^ ^ ^' f -r ir \1 



e 



and (5) becomes 



Vr = ^ e~^ , (6) 



Vkt 



where A is an arbitrary constant. This equation says that Vr is 

 initially indeterminate (evidently infinite, from physical considera- 

 tions) at the centre and zero elsewhere ; as time goes on the value 

 of Vr falls off indefinitely at the centre, rises to a maximum at all 

 other points and then falls off indefinitely also. Now these are 

 exactly the conditions we want for V itself for the case of an 

 infinitely hot point cooling in an infinite medium initially of zero 

 temperature. If we had been studying (3) we would have found 

 the same equation as (6), with x for r and Ffor Vr, for an infinitely 

 hot plane cooling in a medium initially zero. The form of the 

 curves for Fr given by (6) is exhibited on Plates XXIII and XXIV ; 

 with values of r as abscissae curves A^ to A^ are for values of the 



time ^, i , ^^ and ^^ respectively j with values of Ut as abscissae 



curves B"" to B'^ are for values of the distance 0, i, \, | and 1 

 respectively. 



We have taken the form (2) of the differential equation in 

 preference to (1) on account of its symmetry and because we are 

 solving the case of the infinite plane at the same time; but it 

 possesses another important advantage. Since either form of the 

 equation is a linear partial one we can add any number of solutions 

 for a new solution ; the question arises, therefore, whether F being 



a solution -|^and / Vdr are solutions, and what are their physical 



