138 Mr. A. E,. McLeod on the Lags of 



2. Lag J or a Cylinder whose Surface Temperature is Variable, 

 the Surface Conductivity being Infinite. 



Taking cylindrical coordinates, and neglecting the effects 

 of the ends of the cylinder, whose length is supposed great 

 in comparison with its diameter, the temperature u is a 

 function only of r, the distance from the axis of the 

 cylinder. The equation for u is then 



1 ~du _ 1 3» y_M ,_. 



a 2 '~dt ~ r^r ?> 2 ^ } 



Writing u = Re~ a2a % t , the equation reduces to 



dr'^r di> n 



Thus R need be a function of r only, and we have the 

 following particular solution of (7) in terms of the Bessel 

 Function of order zero : 



u — e~ a2a ^ JoOvO- 



The general solution of this type is thus 



u = J A^-^^JoKr). ... (8) 



We must determine the constants A so that (8) shall 

 satisfy the conditions 



u = for t = 0, 

 u r= <p(t) for r = c. 

 We know that 



- 2J (« J> r/c) f* 



«=i riKi) Jo 



where « n is the nth root of J (a)=0. 



Hence, since the value of the integral is a^J^aJ, 





n=l 



and so 



i-2 



2 Jo fcr/c) 



£ c2 



vanishes for £ = 0, and satisfies the differential equation (7). 



