Rectilinear Drawing 69
√ ∕*3π d f- .
— ⅛ / ɛɪɪɪ = -J sιn w)^w∙
Thus the equation under consideration is equivalent to
ξʃ Hι(r')dr = ʃ2Λw (r sin u)du,
at least if the function Я1(г) is suitably restricted near r =0.
For example, it would suffice if we assumed that Hi(r} re-
mains finite near r=0. But the right-hand member is
evidently the same as
/■ ⅛1w (ʃ)- ≠-,
J Vri-Si
so that the equation is essentially a linear integral equation of
Abel type for hχ'∖s}. Hence we obtain
(ʃ) =
i d
4τr 2r
's r{fraHV1')drι')dr
∙∖Λ2 — r2
where the outer integral in the right-hand member clearly
vanishes for r=0 and has a continuous first derivative.
Hence this equation is equivalent to
, ; ч _ i di
hl^~2π dsi
(JraHV1)df1')dr~ _
V si-ri -
Thus we infer that a necessary and sufficient condition for
a solution to exist is that the integral on the right side of the
equation just written admits a continuous second derivative,
in which case the unique solution is provided by the same
equation.
But the double integral on the right may clearly be
written in inverse order of integration as
о ∖J∣Vj2-f2/ Jo