Lags of Thermometers. 51 



It is easily seen that this ratio must occur as a factor in 

 N ; for the surface conductivity term only takes account 

 of the passage of heat across the surface, and if the area is 

 doubled, the heat is lost or gained twice as rapidly and so 

 the lag is halved ; while if the volume is doubled, there is 

 twice as much heat to be lost or gained and so the lag 

 is doubled. Any other non-dimensional factor must be 

 unity, since N = c/3 for a sphere, and c/2 for a cylinder, 

 c being the radius. 



2. Case of a Cylindrical Bulb. (Infinite Length.) 

 If a 2 = K./pcr, we require a solution of 



I ^u __ 1 ~du ~d 2 u , . 



«?a«"r §r §1*' W 



which satisfies the conditions 



u=F(r) for t = 0, (5) 



T§r +tt= *® f ° r r=C ' ^ 



■c being the radius of the cylinder, u the temperature, and 

 t the time. 



The solution satisfying condition (6) is given by Fourier 

 in his Theorie Analytique de la Chaleur, p. 309. The fol- 

 lowing treatment is set out here, however, as it is much 

 more concise than Fourier's * and follows different lines. 

 The general solution of the form required is 



00 



u = ^A n J (a n r/c)e- a Si^\ .... (7) 



n=l 



where A n and a n are arbitrary constants to be determined by 

 the conditions, and J is the Bessel function of zero order. 



The basis of the solutions is the well-known expansion 

 (cf. Whittaker, ' Modern Analysis,' p. 374, Ex. 20) 



, , oo f V(f)JoGWf 



Jo 

 where 



AJo'(/3„) + HJ (A) = 0, B.=chjK. . . (9) 



* See also, Carslaw, ' Introduction to the Theory of Fourier's Series 

 .and Integrals,' p. 208. 



E 2 



