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Full text of "Scientific Papers - Vi"

1911]                    PROBLEMS IN THE CONDUCTION OF HEAT                           53
In Kelvin's summary linear sources are passed over. If an instantaneous source be uniformly distributed along the axis of z, so that the rate per unit length is q, we obtain at once by integration from (2)
g dz e
qe
.(8)
From this we may deduce the effect of an instantaneous source uniformly distributed over a circular cylinder whose axis is parallel to z, the superficial density being or. Considering the cross-section through Q—the point where v is to be estimated, let 0 be the centre and a the radius of the circle. Then if P be a point on the circle, OP = a,OQ = r, PQ = p, z POQ = 9; and
p* = a2 + r2 — 2ar cos 6,
so that
4
T
.(9)
JT0 (so), equal to J0 (ix), being the function usually so denoted. From (9) we fall back on (8) if we put a = 0, ITTCLO- = q. It holds good whether r be greater or less than a.
When ao is very great and positive,
In 0») =
.(10)
so that for very small values of t (9) assumes the form
(r-g)«
_       era,         —^— V ~ 2 V(TrraS) 6
vanishing when t — 0, unless r = a.
Again, suppose that the instantaneous source is uniformly distributed over the circle £ = 0, f = a cos 0,77 = a sin 0, the rate per unit of arc being q, and that v is required at the point a), 0, #. There is evidently no loss of generality in supposing y = 0. We obtain at once from (2)
.(11)
where
Thus
= a? + a? + 2 - 2ax cos
qa
.(12)
from which if we write q = crdz, and integrate with respect to z from — oo to + oo , we may recover (9).(9) are of the standard form if we take