Okr — Stability or Instability of Motions of a Viscous Liquid. 95 



CHAPTEE II. 



The Fundamental Feee Distuebances of a Steeam which is Sheaeing 



Unifoemly. 



Aet. 13. The Feriod-JEquation for the Buundary- Conditions V^v = 0. 



In a passage quoted above,* Lord Eayleigli appears to suggest that possibly 

 in the case of a stream of uniform vorticity there may not be free disturbances 

 which involve the time in the usual exponential or trigonometrical form, i.e. 

 varying as e^^ where q^ is a real or complex constant. I proceed to consider 

 this question. Eeferring to Lord Kelvin's analysis given in Chapter I., if in 

 equation (20) of that Chapter, we write oji = qj, it assumes the form 



d'Sldif = {P + n~ + ( 2^ + ilMh ]8, (1) 



where 



;S' = [d}\df - P - 1V) V. 



(2) 



The solutions of (1) are given by Lord Kelvin in the form of infinite series; 

 and the equation had previously been discussed by Stokesf and others. The 

 solution in fact is, if we replace l^ + ii? by A^ 



S=[vX'+qj + liiyi)h 



All 



2 |/g 

 31 V 



■y% 





+BI-1 



•2(53 

 3 V 



■yi 



v\^ +qj 



;3) 



where /„ is the function connected with the Bessel function t/„ by the relation 



In{6)=i-'^Jn{id) = - 



1 + 



2'*n {n) \ 2 . {2n +2) 2.4. (2?t + 2) (2??, + 4) 

 We may also write (3) in the form 



where 



+(^=^+374+3.4.6.7 



: + 



4>(Y) = 1 



ys 



ye 



2.3 "^ 2.3.5.6 ^" 



(4) 



(5) 



(6) 



(7) 



* See Art. 5, p. 85. 



t It was in connexion \yith Ihi^i equation that Stokes published liis investigation of tlie asymptotic- 

 expansion of Bessel's functions ; " On the Numerical Calculation of Definite Integrals and Intiuite 

 Series," Trans. Oamh. Phil. Soc, ix., Part i., 1850 ; Math, and Phys. Paper ii, p. 329. 



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