6 Flow of Compressible Fluid past an Obstacle. 



approximation, but this character is preserved however far 

 we may continue the approximations. And since the co- 

 efficients of the various P's are simple polynomials in p, the 

 integrations present no difficulty in principle. 



Thus far we have limited ourselves to Boyle's law, but it 

 may be of interest to make extension to the general adiabatic 

 law, of which Boyle's is a particular case. We have now to 

 suppose 



P/Po=(p/po) y , (25) 



making 



trW"-<r- ■ ■ ■ » 



if a denote the velocity of sound corresponding to p . Then 



by (3) 



£sr--* (», 



If we suppose that p corresponds to <7 = 0, C = a 2 /(y— 1), 

 and 



,er- i - te ij* ! w 



whence 



d\ogp_ dq 2 /dx 



dx ~ 2a 2 -(y-l)q 2 ' ' ' ' K -> 



The use of this in (2) now gives 



T 2a 2 —(y—l)q 2 Idx dx dy dy dz dz J v ' 



from which we can fall back upon (6) by supposing 7=1. 

 So far as the first and second approximations, the substitu- 

 tion 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 synchro- 

 nizes witli 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. Koy. Soc. A. vol. lxxxiv. p. 247 (1910) ; ' Scientific Papers/ 

 vol. v. p. 608. 



