402 Professor Carslaw, The cooling of a solid sphere 



diagram, joining the ends of these two curves. Then use Cauchy's 

 Theorem and we see that U2, = hva when r = h. 



Thus the condition (2') is satisfied. 



There remains only the initial condition, u^ and u^ are to 

 vanish when ^ = 0. 



Now the equation 



F {a) = a- cos aa sin /xa {b — a) + sin aa cos fxa {b — a) 



sinaasin |U,a (6 — a) = ...(9) 



+ 



fxaa 



Fig. 2. 



has no imaginary roots, or repeated roots, and it has an infinite 

 number of real roots symmetrically placed with regard to the 

 origin*. 



Putting ^ = in (6) and (8), we have the integrals 

 bV(f f sin ar da , bv^ f G (a) da 

 ~u7Tj F{aj''a ^^ tTT j F{a) ~a ' 



Consider the closed circuit of Fig. 3, consisting of the path (P) 

 and the part of a circle, centre at the origin, lying above the path 

 (P). There are no poles of these integrands within this circuit and, 

 when the radius of the circle tends to infinity, the integral over 

 the circular arc vanishes. 



Fig. 3. 



It follows that both integrals vanish over the path (P) and the 

 initial conditions (3) and (3') are satisfied. 



* See below § 5. 



