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Rectilinear Drawing          67

which we write in the form

fo'⅛(r sin χ)(cos χ cos χ - sin χ sin χ')dχ=0.

On integrating by parts the first term evidently takes the
form

1 [

-   ∕2w (r sin χ) sin χdχ,

rJo

while the second is seen to be equal to
d f

-jf I fz'^'' (r sɪn χ) sin χdχ.

Hence if we write

W = X*7?0 (r sin χ) sin χdχ,

we conclude that

1jγ-jγ≡o,

r

i.e., W = cτ. But since /Fz(0) =0 we must have c =0, and so
F≡0.

Employing now the same argument as in the case m = 1,
for the equation
W = Owe find that f∙ι's) =0 and so ∕a(j,)≡0.

Likewise we readily infer that gs(∕)≡O.

More generally, we may obtain the stated result for
m = 1, 2 ■ ∙ ∙ in succession, by writing at the mth stage

cos =cos χ cos (tn — l)χ —sin χ sin (m-l')χ
sin =sin χ cos (m — l)χ+ cos χ sin (m-l)χ

in the equations involving ∕m(j) and gm(j∙) respectively, and
thus reducing the question to one of the type considered at
the (tn —l)st stage.

In this manner the stated result is readily established.

We are now in a position to formulate preliminary neces-
sary and sufficient conditions for a continuous solution of the
general harmonic case of the
mth order.



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