1916] ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE ' 405
c denoting the radius of the sphere. As in (8),
Qft^G Q/-B\ — — — ,_, — 4- ] P
doc dx dr dr ~*~ r2 dO d9 ^ 1V 5?'7 2r10/ x
on substitution from (16) of the values of 0 and cf. This gives us the right-hand member of (6).
In the present problem
J»- ' r dr'1 r'sinfl'^^1'^" ...............(18)
while Pw satisfies
-a~ sm 6 do
(19)
x '
so that V2</> = ^Pn ................................. (20)
i ± $4> 2 d<h n(n + 1) . 01T, /01S
reduces to -y-if H --- ,-i -- ^ — - — ' <b = r?Pn ...................... (21)
dr" r dr r* r v 7
The solution, corresponding to the various terms of (17), is thus
With use of (22), (6) gives
°°P 3c8p« 8t!"p3 3c°P
- _ ,
"
5r5 24r8 10r2
+ ^rPa + ir-sP: + Cr»P, + Dr-"P8) ............... (23)
A, B, C, D being arbitrary constants. The conditions at infinity require A — U, 0 = 0. The conditions at the surface of the sphere give
and thus </> is completely determined to the second approximation.
The P's which occur in (23) are of odd order, and are polynomials in fj, (= cos 6} of odd degree. Thus dfyfdr is odd (in /i) and d<p/d6 = sindx even function of //,. Further,
(f — even function + sin2 6 x even function = even function, dq^/dr = even function, d<fjdO = sin 6 x odd function.
Accordingly
dd> do2 1 d<k> dq2 ,, . .. r j j + 1 ja j^ = odc"- function of ^, ar rf?^ r2 a# rt^ ^
and can be resolved into a series of P's of odd order. Thus not only is there no resultant force discovered in the second approximation, but this characterme lines with use of Legendre's functions Pn (cos 0) in place of cosines of multiples of 0. In terms of the usual polar coordinates (r, 6, ««), the last of which does not appear, the first approximation, as for an incompressible fluid, is