408 Professor Garslaiv, The cooling of a solid sphere 



In the first place we confine ourselves to the case when 

 11 (b — a)/a is not rational. 



As in § 5, let a, ^ be two different positive roots of (28). 



Also let 



Ui = sin /xa (6 — a) sin ar, < r < a] 



U 2 — sin aa sin jjia {b — r), a<r<h\ 



And let 7^ , V^ be the corresponding expressions when ^ is sub- 

 stituted for a. 



We know from § 5 that 



^''u^V^dr + ^\^ U^y^df^^) (30). 



Following Fourier's method, our solution is obtained by ex- 

 panding VqT in an infinite series of these terms : 



VqT = S ^„ sin ^a„ (6 - a) sin «„r, < r < a) 



I ..,(31), 

 = T^ An sin a^a sin jua„ {b — r), a< r< b\ 



n ) 



Then mth the usual assumptions as to the possibihty of this 

 expansion and of term by term integration of the series, we have 

 from (30) 



sin2 ^a^ {b~a)l sin^ a^rdr + ^ sin^ a^a f sin^ ^a„ (6 -r)dr ■ 



Jo cr J a J 



sm fjMn {b — a) I r sin a^rdr 



Jo 



a . fb 



-\ — sm a„a r sm ^a^ (6 — r)dr . 



^ J a 



Evaluating these integrals and using (28), it will be found that 



A 2hvrf sin a„a 



^" ' ^T r--(32). 



= V 



a„ I a<T sinVa„(6 -a)+ix{h-a) sin2a a + -^- sin^a„a sm2 ua„ (h - a] 



Also, from (26), it will be seen that the solution of the problem is 



„, _ o;,,, ^ sin a„a sin ^a„ {h - a) sin a„ r , e-Kia,ri 



"i-^wo^-— ^ __ £ : ...(33), 



acrsinVa„(&-a)+/i(6-a)sin^a„a+ ^sin2a„asinVa„(^-«) "" 



«2 = 2;>t'o2 sin2 a^a sin ^g., (?)->•) e-K^o.nH 



ao- sin2;xa„ (6 _ a) + ^{6 - a) sin2a„a + ilji^sin^a^asin^^a,, (b - «) "" 



This agrees with the results given in §2 (10) and (11), if we 

 remember that u^ and u^ now correspond to [rv^ - u,) and [rvr, - u.) 

 of that section. 



