in the theory of Conduction of Heat 415 



which yield the formulae 



„ Aa sinh qr „ Aa sinh q^ {b — r) b sinh q^ (r — a) 

 r sinh qa ' ^ r sinh q^ I r sinh q^ I ' 



because V = Vi = A at r ^ a, and Fj = 1 at /• = 6. 

 Further we are to have 



or 9r 



and so 



kA [q coth qa ) = — Ic.A (q-. coth qd + - ) H —^ — -, , 



V a/ \ «/ asinh^iZ 



• * ^ Oift/sinhoiZ 



giving* A = ^^'^^ — 



® (A;/^i) (ya coth g'a — 1) + (g'lCt coth g-jZ^- 1) ■ 



If we write kl/kjb = s, this formula can be written 



^ gi?/sinhgi? 



5 {qa coth g-a — 1) + (a/6) (g-jZ coth g^Z + l/a) ' 

 The problem actually amounts to evaluating 



l^^-^ atr = 6, 



Vq or 



where g is the gradient of temperature at the outer surface of the 

 shell at time t. It will be found at once that 



An expression for ^^ as a Fourier-sum can be found by Heaviside's 

 general process; but as this should be equivalent to the result 

 calculated by Prof. Carslaw, I do not stop to write out the result 

 and proceed to simplify the above formulae by means of approxi- 

 mations suitable to the data of this particular problem. These 

 approximations correspond to (i) treating qa as having a sufficiently 

 large real part to allow coth qa to be replaced by unity and 

 (ii) treating q-J, as small. 



The above formulae reduce then to the following, on rejecting 

 iqil)* and higher powers, 



, _ 1 - i iqii)' 



l-s + sqa+^, (a/b) {q,l)^ ' 



* On writing q=i.a, qi=i/j.a, k/k^^fxcr, it will be found that 



Aa — h sin aa/F (a), 

 where F (a) is the function defined in Prof. Carslaw's paper. Then Fj is easily 

 seen to lead to formula (7) of that paper. 



VOL XX. PART IV. 27 



