Oer — Stability or Instability of Motions of a Viscous Liquid. 101 



or, as it may be written, 



- 'i(e«i-«2 _ c"2-"i) + e-«i-«2 ^ 0. (23) 



Substituting 



Wi = P + iQ, u^^P ~ iQ, (24) 



tbis becomes 



2 sin 2Q + e'^"" -= 0. (25) 



Moreover, tbe form of (8) sbows that when P is real, the accurate vahie of 

 the left-hand member of (23) is a real quantity; and (10), (11) show that 

 the errors in the expressions e"i , e""i have moduli less than those of Auf'^ e^\ , 

 £ui~^ c-"i , respectively ; and those in e"2 , e""^ have moduli less than those of 

 Ai(,2-^e"2, Bu2-^e~"2^ respectively, where A, B are certain numbers. Thus the 

 error in the left-hand member of (25) is less than 



2\{1 + AU-') {I + BU-') -1] +&-'-'' [[1 + BU-f-- I], (26) 



where U denotes the modulus of u^ or U2. And if 



is large enough, P, U can be made as great as ever we please. From this 

 it is evident that, if 



v\ l^ J 



is sufficiently great, on substituting a real value oip in the accurate expression 

 for the left-hand member of (25), there is obtained a real magnitude which 

 differs from 2 sin 2 Q by as little as ever we please. Consequently, for all 

 values of /, X, there are an infinite number of real negative values of ^3, given 

 as nearly as we please by the equation 2Q = rir, where r is a large enough 

 integer. 



Art. 17. For Waves of Sufficient Length in the direction offloiu, all Disturbances 

 are AiJerioclie, the values of p being given aijproximcdcly by eciuation 

 (15). 



The period-equation may be written in the form 



2«y 2«''(21^'2 + ^2/2^^2) 4ay(9^'2 + jS^/V) 



1 + — — + 77z~z, — ;; + 



315v^- 2835 v^ 



2a« 429j/^ + 78j/-j3-/Vt^ + ^Hht' 4ft'y (117/^ + SO/^/J'^/Vr + jS'ZV^^) 

 ^ 1216215.;* ^ 18243225v^ + . . . - 0, 



(27) 

 where -p' = p + v\^, and accordingly if [5la^/v is small enough, it is evident 



R. I. A. PROC, VOL. XXVII., SECT. A. [14] 



