Ok'R — Stabilitij or Instability of Motions of a Perfect Liquid. 55 



first the former of these integrals, at the time in question the angle whose 

 cosine and sine occur does not involve p. 



Suppose that Ic (p - h) is large. It is not difficult to prove that when this 

 is the case 



and that /i (kh) 



p"/i (kp) dp = T'^I, (At) K, (kb)/Jc, 



b 



p" Ki (kp) dp is negligible in comparison. 



Thus, k (r - h) and mr being large, if we further suppose that ni^r/k is 

 large, the only term to be taken into account is that involving m*(0^ and the 

 approximate value of the first term of (65) is accordingly 



2m^rVri /i (kr) K, {J^h) cos {kz - oii^b^ - kC't). (66) 



It may next be proved that, at the time in question, the second term of 

 (65) is very small in comparison with (66), provided, of course, that the 

 cosine which occurs in the latter is not a very small fraction. In this second 

 term consider first the portion involving m^p^, i.e. : 



2m' f ' p' cos{2m>- - mW- - kz + liC't). \_I, {kp) K, (kb) - 1, H^b) K, [kp)] dp. 

 J b 



(67) 

 On integration by parts this may be written 



Imh^'' [/, [kr) K, [kb) - /, {/(b) K, (At)] sin {27/i^r' - m^^^ - kz + kC't) 



- W [' sin { 2mV^ - iri'b'' - kz + liC't] .'j-[p'{ I, (Icp) K, {kb) - 1, {kb) K, {kp) ] ] dp. 

 Jh dp 



(68) 



The first term bears to (66) a ratio of order kjm^r. As regards the second term, 

 the differential coefficient which appears in it is positive throughout the 

 range, and consequently the integral is less, and, as a matter of fact, much 

 less, than if the sine were replaced by unity, in which case it would be of the 

 same order as the first term. Thus, the portion of the second term of (65) 

 which involves m'p'^ is negligible. 



As regards the other portions of the second term of (65), each is, since 

 I,{kp) K,{kb) - I,{kb)K,{kp) 



is positive, less than if the cosine or sine were replaced by unity, and even 

 then they would be negligible in comparison with (66). 



And this argument applies when b is zero, for division by the infinite 

 iTi {kb) which then occurs in the left-hand member of (29), eliminates 

 any disturbing infinity. 



Thus, in (65), only the first term need be taken into account, and its 

 approximate value is given by (66). This, then, is to be substituted for the 



