XVI
267
FLOATING ELASTIC PLATES
12(1 - μ²)s
Eh3
2
= a*:
=
1
a
>
our problem may be stated mathematically thus: Required
an integral of the equation *z+a⭑z = 0, which vanishes at
infinity, is finite at the origin, and is such that the integral of
sa*f*zdw taken throughout the neighbourhood of the origin
may be equal to P.
With Heine¹ we write
eip cos in
du,
K(p) = [e's
=
then K(p) is a solution of the equation
K(pa¹) is a solution of our equation. And
2=
a²P
Απεί
²zz, and therefore
{K[ap √}(1 + i)] − K[ap√(1 − 1) (1)
-
is also a solution. It is real, and if we bring it into a real
form we get by transforming the integral
∞
a²P
€
2=
4πS
1
-ap√v sin ap √‡vdv
√o² 1
(2),
-
which form shows that the solution assumed vanishes at
infinity. In order to examine its value near the origin, we
employ an expansion of the function K given by H. Weber,2
first in a series of Bessel's functions, then of powers of
and thus obtain
P,
2=
a²P (a²p²
2πS 22
log ap-
αδρά
22.42.62
(log ap-)+..
T
a*p*
+ 1
-
+
asps
22.42 22.42.62.82
-...)
(3).
-(1+ log 2 - C)
22 αθρό
22.42.62
+
¹ Heine, Handbuch der Kugelfunktionen, vol. i. p. 192, 1878.
2 l.c. p. 244. The sign of C is wrongly quoted here.