with a concentric core of a different material 403 



Finally the solutions in (6) and (8) are obtained as infinite 

 series. For we have 



_ 6^0 r sin ar e-*'"'^^ , 

 ""^^ 2177.1 #(aT~^ 



and ^2 = ?r^ I rTTTr! — r~ ^«' 



6^0 [G (a) e^"^ 



over the path {Q) of Fig. 2. 



Then, by Cauchy's Theorem, we have 



^, ^ sin a„f e""'''"'' .,„> 



1 -t' VO^wj «n 



(rCOsa,jrtsm^ci,^(r- a) + sina,,rtCOS;tta„(/'-a) ^ sma„rt sin/xa,((r -a) -^^^LnH 



n'o + 2broS • — — ; 



(11). 



the summation being taken over the positive roots of (9). 



3. In the discussion of § 2, put V-^ = Vq~ v-^ and Fg = Vq ~ '^2- 

 Then V^ and Fg are the temperatures in the core and sur- 

 rounding sphere, when the surface r = 6 is kept at temperature 

 zero and the initial temperature of the whole is Vq. 



Also the gradient, when r = b, namely -^ , is given by the 



equation 



a■cosa,^acosfxa,^{b - a) H ^sina„flCOSjua„(& - a) - siiia,/! sin/xa„(?^ -a) 



),.r ' ""■'■ ! """ ^'>,.) -' 



the summation extendmg over the positive roots of (9). 



4. With the constants which Perry and Heaviside adopted, 



a = 6-38 X 108, 6 - a = 4 X 10», Vq = ^ x 10^ 

 K^ = -47, K^ - -00595, 



CiPi-2-86, C2p., = -507, 



Kj - ^i/ci/oi = -1643, K., = AVC2P2 = '0117*, 

 ju = VC'^i/'^-i) = 3-742, a = ViKiC^pJE^c^^) - 2M. 



Thus ^cT= 79 and /x ^^^^ - 2-35 x lO-^. 



Also the gradient of 1° in 50 ft. is 1/2743 degrees per cm. 



* This corresponds to Kelvin's value of 400 for k in foot-year units (loc. cit. § 15 

 and Mathematical and Physical Papers, vol. 3, p. 302). 



..(12), 



