1902.] MACKENZIE — EQUATIONS OF HEAT PROPAGATION. 195 



from the point P, at which we want to know the temperature, 

 meets this plane. Call the length of this perpendicular x. Break up 

 the lamina into concentric rings of radius p about this origin, and 

 let the distance of every point in one of such rings from the point 

 P\>^ r and the thickness of the lamina Ax\ then we have 



8( 



_ - V- + p- f -_ 



— ^, I e ^r.p. Ax. dp = — 5 — . g /30) 



From the symmetry of the problem this is evidently a case of linear 



flow, and the solution must satisfy equation (3). Knowing this 



solution (we can get it otherwise), the solution for three dimensions 



given in (15) can be deduced ; we have but to multiply the value of 



/>' 

 -T7 for the case of one dimension by two similar expressions with 



y and z respectively substituted for x. 



The corresponding electrical problem is that of an infinite cable 

 with no lateral loss by leakage touched for an instant to a condenser 

 of potential V^. If there is lateral leakage equation (20) is still 

 the solution of the electrical problem; Vis then not the potential, 

 but the potential can be derived easily from it, as is well known. 



If Q or (7, according to the unit of heat used, is the amount of 

 heat required to raise the mass of a section of the plate of unit 

 area by V^ degrees, then Q = CDVqJx, or c- = V^Ax, and equa- 

 tion (20) becomes 



Q -Ul ^ --Ikt 



V = —-^ ^ e = ^ e (21) 



Of course this equation is of only the same grade of approximation 

 as (15). It will be the more nearly exact the smaller Ax and, since 

 the product of V^ and Ax measures the heat in a section of unit 

 area and is to remain constant, the greater V^. In the limit we 

 should have the solution for an infinitely hot plane. The form of 

 this solution we have already found ; it is from (G) and the remarks 

 following it 



A 



ikt 



,-=.e (22) 



Calling Q the total heat associated initially with a unit of area of 



