Gases under Hiyh Pressures. 183 



The above calculation of X takes account only of the 

 principal vibration. Other vibrations are possible corre- 

 sponding to higher roots of (10), and if these occur appre- 

 ciably, strict periodicity is lost. Further, if we abandon the 

 restriction to symmetry, a new term, r~ 2 d 2 $[d6 2 , enters in 

 (13) and the solution involves a new factor cos (w0 + e) 

 in conjunction with the Bessel's function J n in place 

 of J . 



The particular form of the differential equation exhibited 

 in (13) is appropriate only when the section of the stream is 

 circular. In general we have 





/* 2 )4> = 0, 



(16) 



the same equation as governs the vibrations of a stretched 

 membrane ('Theory of Sound,' §194). For example, in 

 the case of a square section of side b, we have 



■ 



<£= cot 



. cot 



7T2/ 



'JL p i(kat+pz) 



(17) 



vanishing when x=.+^b and when y=+^b. This re- 

 presents the principal vibration, corresponding to the 

 gravest tone of a membrane. The differential equation is 

 satisfied provided 



/c 2 -/3 2 = 27r 2 /^ 



as) 



the equation which replaces (15). It is shown in ' Theory 

 of Sound ' that provided the deviation from the circular 

 form is not great the question is mainly one of the area of 

 the section. Thus the difference between (15) and (18) is 

 but moderate when we suppose 7rR 2 equal to b 2 . 



It may be worth remarking that when V the wave- velocity 

 exceeds a ) the group-velocitv U falls short of a. Thus in 

 (17), (18) 



V = 



ha 

 J' 



U 



_ tfQQV) _ dk___IBa m 



d/3 



djd k 



so that 



UV = a 3 . 



(19) 



Returning to the jet of circular section, we may establish 

 the connexion between the variable pressure along the axis 

 and the amount of the swellings observed to take place 



fffl 



