f 



2 Lord Rayleigh on the Flow of 



steady motion is in the usual notation 



dx dy dz ^ \dx dy dz J ' 

 or, if there be a velocity-potential <j>, 



d<j> dlogp + d$ dlogp + <ty <Hog/j + «^ _ q > / 2 ) 

 d# d# rf?/ dy dz dz ™ ~ 



In most cases we may regard the pressure p as a given 

 function of the density p, dependent upon the nature of the 

 fluid. The simplest is that of Boyle's law where p = a 2 /o, 

 a being the velocity of sound. The general equation 



^=G~W, (3) 



where q is the resultant velocity, so that 



q 2 = (d<f>/dxy + (d<l>/dyy + (d<t>/dz) 2 ... (4) 



reduces in this case to 



a 2 \ogp = C-iq 2 , 

 or 



a*log(p/p )=-tf 9 (5) 



if /o correspond to 2 = 0. From (2) and (5) we get 



2a 2 \dx dx dy dy dz dz J' 



When q 2 is small in comparison with a 2 , this equation may 

 be employed to estimate the effects of compressibility. 

 Taking a known solution for an incompressible fluid, we 

 calculate the value of the right-hand member and by in- 

 tegration obtain a second approximation to the solution in 

 the actual case. The operation may be repeated, and if the 

 integrations can be effected, we obtain a solution in series 

 proceeding by descending powers of a 2 . It may be pre- 

 sumed that this series will be convergent so long as q* is 

 less than a 2 . 



There is no difficulty in the first steps for obstacles in the 

 form of spheres or cylinders, and I will detail especially 

 the treatment in the latter case. If U, parallel to = 0, 

 denote the uniform velocity of the stream at a distance, the 

 velocity-potential for the motion of incompressible fluid is 

 known to be 



<£ = U(r + c 2 /r)cos0, (7) 



the origin of polar coordinates (r, 6) being at the centre 



