The name is absent

Rectilinear Drawing          75

If we multiply through by if^m+' and integrate from 0 to
r, there results

(1 ) fr^m-^,y ≈ ʃ h_m^'∖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 h_rJ'^m'^j (r sin u)du.

If now we replace the wι-fold integral on the left by a
simple integral, (4) becomes

/ A^m ∣∙_r                      ∣^,aτ

(5)    2 ∖2∕ ff_m(t)(r^s-t^s)^m-'dt= k_m<-^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 A_m(∙r) as thus defined exists and is continuous,
the equation (1) will be satisfied by λ_m(j).

But the right-hand integral, after inverting the order of
integration, becomes

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