406 ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE [407
is preserved however far we may continue the approximations. And since the coefficients of the various P's are simple polynomials in 1/r, the integrations present no difficulty in principle.
Thus far we have limited ourselves to Boyle's law, but it may he of interest to make extension to the general adiabatic law, of which Boyle's is a particular case. We have now to suppose
.............................. (25)
making f = ^ (^T = a- (^ , ..................... (26)
6 dp p0 \pj \pj v '
if a denote the velocity of sound corresponding to p0. Then by (3)
C-tf ......................... (27)
** v '
7- p If we suppose that pQ corresponds to q=> 0, (7= n2/(7 — 1), and
, dlogp . ,
whence — -s_r=_ — - — u. — — — ...................... (29)
dx 222 v
The use of this in (2) now gives
(dx dxdy dy dz dz '
from which we can fall back upon (6) by supposing 7 = !. So far as the first and second approximations, the substitution of (30) for (6) makes no difference at all.
As regards the general question it would appear that so long as the series are convergent there can be no resistance and no wake as the result of compressibility. But when the velocity U of the stream exceeds that of sound, the system of velocities in front of the obstacle expressed by our equations cannot be maintained, as they would be at once swept away down stream. It may be presumed that the passage from the one state of affairs to the other synchronizes with a failure of convergency. For a discussion of what happens when the velocity of sound is exceeded, reference may be made to a former paper*.
* Proc. Roy. Soc. A, Vol. LXXXIV. p. 247 (1910) ; Scientific Papers, Vol. v. p. 608. [1917. See P.S. to Art. 411 for a reference to the work of Prof. Cisotti.], as for an incompressible fluid, is