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1911]                      PROBLEMS  IN  THE  CONDUCTION  OF  HEAT                              63
which, if we take r \/{ip) = z, as before, may be written
-7 -,3                               /2                   V         t                 y  ".....*y        /
\jjfij                            z
and is of the same form as (69) when in the latter n   is written for n.
As appears at once from. (80), the solution for the interior of the cylinder may be expressed
v = A cos nO e*' Jn (i:/2pl/'2 r),   .............'........(82)
Jn being as usual the Bessel's function of the nth. order. For the exterior we have from (81)
A = B cos n6 eW err^(*'/n _ ^ (i*^* r),   ...............(83)
- 12) (4n'-3)(4n-5)
-- + ............. (84)
The series (84), unlike (77), does not terminate.    It is ultimately divergent, but may be employed for computation when z is moderately great.
In these periodic solutions the sources distributed over the plane, sphere, or cylinder are supposed to have been in operation for so long a time that any antecedent distribution of temperature throughout the medium is wibh-out influence. By Fourier's theorem this procedure may be generalised. Whatever be the character of the sources with respect to time, it may bo resolved into simple periodic terms ; and if the character be known through the whole of past time, the solution so obtained is unambiguous. The same conclusion follows if, instead of the magnitude of the sources, the temperature at the surfaces in question be known through past time.
An important particular case is when the character of the function is such that the superficial value, having been constant (zero) for an infinite time, is suddenly raised to another value, say unity, and so maintained. The Fourier expression for such a function is
a'mpt . p     P
the definite integral being independent of the arithmetical value of t, but changing sign when t passes through 0 ; or, on the understanding that only the real part is to be retained,
fl* .
^ ............................... (86)......................... (80)