1^0 Proceedings of the Rojjal LHsii Academi). 



There is obtainable as a special and limiting case the solution of the 

 problem of the free disturbances of the fluid at rest ; these have been inves- 

 tigated by Lord Eayleigh.* In this case, |3 being zero, if p' is finite u^, % are 

 infinite, but Ui-u~, or 2cti{- 2^'lvf^ is finite; if, in (72), in the first and third 

 expressions in { j , we neglect all terms which do not involve Exp f+ u), and 

 then equate /3 to zero, we obtain an equation which is valid and exact 

 over all the plane ; it may easily be verified that this equation leads to 

 Lord Eayleigh's results. 



Another special case which may be noticed is that in which \a is very 

 great. In this case the smaller roots, i.e. those for which X is very much 

 greater than {{-]_)' + lj5ai)fv } i, are given approximately by the same formulae 

 as when the boundary-conditions include S = ; and for those which are not 

 so given ^j' is wholly real and negative. In fact, for those real values of ^j' 

 which are far removed from the complex ones, the equation assumes the 

 approximate fornr 



■ [A - * { (- 1/ + li5ai)lv ) =] [X - ^■{ (- p' - l^ai)lv ] =] 



_^ - X^ + v-\i3^ + Z^/3Vi^)-^ - Air^ ! - 2j/ + 2 iiP + I'^hi'f]^ 



^ -\' + v-\v" + P\^crf + iXin [ - 2p' + 2(p'^ + r-iiht'f j 4 ' ^ ^ 



This equation could be solved without any great difficulty if the values of 

 the constants were given. It will be seen that in taking successive values of 

 p in order of increasing magnitude, in passing through the region in which 

 p"^ and v\^ are of tire same order, one root is, so to speak, lost as compared 

 with the period-equation (8). All the roots of the equation (72) are thus 

 accounted for. 



In the most general case, the real values of ^ which are not too near the 

 complex ones are given by i 74). As regards the determination of the complex 

 values, thougli (72) simplifies somewhat, I have not been able to reduce it to 

 a form which I can solve. 



The approximate forms (72), (74), which have been obtained for the period- 

 equation are inappropriate to small values of X«, as when \a is made equal to 

 zero, they become identities ; when \a is very small, it is more convenient to 

 express (70) in the form 



■•a Ca ra 



Si cosh \ydy S2 sinh Xydy - Si sinh Xydy S2 cosh \ydy = 0. 



! J -n J - a J - a 



' (75)^ 



* " On the Question of the Stability of tlie Flow of Fluids," I'hil. Mag. xxxiv., 1892, p. 59 ; 

 Collected Papers, iii., p. 582. 



