Rectilinear Drawing 75
If we multiply through by ifm+' and integrate from 0 to
r, there results
(1 ) frm-,y ≈ ʃ hm^'∖r sin u)e^i''m^'-',udu,
since the right-hand member vanishes to order m at least in
r. But (1') is of a form like (1) on the right except that m
is replaced by m — 1. Repeating the same type of procedure
wi — l times, we obtain finally
w ^∙∙∙m⅜
ʃ hrJ'm'j (r sin u)du.
If now we replace the wι-fold integral on the left by a
simple integral, (4) becomes
/ Am ∣∙r ∣,aτ
(5) 2 ∖2∕ ffm(t)(rs-ts)m-'dt= km<-m>(r sin u)du.
(wi —l)!∙∕o Jo
But hm^m∖r') is even in r so that the integral on the
right in (4) is four times the same integral taken between 0
and τr∕2. Writing ʃ =r sin и we thus obtain, as the equiva-
lent of (4), a linear integral equation of Abel type in
hm^mXs) with explicit solution
h w=ι G) d^
π (wι-l)! ds"+'
Γr( ......- ∙
Uo ∖Jo t k ,
Conversely, if Am(∙r) as thus defined exists and is continuous,
the equation (1) will be satisfied by λm(j).
But the right-hand integral, after inverting the order of
integration, becomes