94 Lord Rayleigh on [}1&J 15, 



motions represented by (12) take place in perfect independence of one 

 another. 



For the free motion we get by Lagrange's method from (16), (22), 



Van T gffl . J/W ( 8 + j&2 j,^ Q ^4), 



dt 2 



which applies withont change to the case n=0. Thus, 

 if otn OC COS (pt— e), 



2 = T il'a Jn (iJc a) ^ + w _ ^ ^ # ^ ^ 

 pa 5 o n (ika) 



giving the frequency of vibration in the cases of stability. If ?i = 0, 

 and Jca< 1, the solution changes its form. If we suppose that «„ oce±2'. 



g2= t _ ^ ^ (2 



/aa d J (2fta) 



From this the table in the text was calculated. 



When 7i is greater than unity, the values of ]fi in (25) are usually 

 in practical cases nearly the same as if ha were zero, or the motion 

 took place in two dimensions. We may therefore advantageously 

 introduce into (25) the supposition that Tea is small. In this way we 

 get 



pa L n . 2n + 2 J 



or, if Jca be neglected altogether, 



P 2 =(^-^)-3 (28), 



T 



pa 1 



which agree with the formulas used in the text. When n=s\, there is 

 no force of restitution for the case of a displacement in two 

 dimensions. 



Combining in the usual way two stationary vibrations, whose 

 phases differ by a quarter of a period, we find as the expression of a 

 progressive wave, 



r=a + 7 a cos nO cos Tcz cos pt-\-y n cos n0 sin Jcz sm.pt 



=a +7« cos kiB cos (pt—Jcz) (29). 



For the application to a jet the progressive wave must be reduced 

 to steady motion by the superposition of a common velocity (v) equal 

 and opposite to that of the wave's propagation. The solution then 

 becomes 



r=a + 7» cos n6 cos l'z .... (30), 



