408 Dr. T. J. I'a. Bromwicb : Examples of 



the present section the formula will be used to obtain the 



results given by Mr. McLeod and some extensions of them — 



found by allowing for surface-conductivity. 

 The new formula may be stated as follows : — 

 Suppose that we have found, by operational methods, the 



symbolic equation 



•*=§f)( G <), a) 



where p stands for the operator d/dt and G is a constant ; 

 then the solution (1) is to be interpreted as 



.-G^+Ha+S^^},. • • (2) 



where p = ot is any root of A(p)= : 0, the summation extends 

 to all such roots, and N , N x are defined by the algebraic 

 expansion 



2^ ==N » +N ^ +N ^ 2+ w 



The solution (2) has the property of reducing to zero at 2 = 0. 

 Heaviside's equation * is given similarly in the form 



a(p) i iNoH r*A» e /' • • • w 



where P is a constant. It will be noticed that equation (4) 

 can be derived from (2) by differentiation with respect to t ; 

 but the reverse step of integrating (4) does not give the 

 constant Ni immediately. We proceed now to apply equation 

 (2) to the heat-problems mentioned. 



(a) The surface of a sphere is maintained at temperature 

 21= Gd. 



The differential equation for u in the sphere can be written 



w 



here a 2 = K/pcr, I 



5? (n, >=a^"*M (5, 



K being the thermal conductivity, p the density, and <r the 

 specific heat of the substance of the sphere. 

 Now writing symbolically p for 'd/'dt and 



f'-p*l*, (6) 



* ' Electrical Papers,' vol. ii. pp. 226 and 373. 



