Orr — Stability or Instability of Motions of a Perfect Liquid. 51 



At times such as considered the first term is of order e^''lm. And the second 

 is evidently less than (and, as a matter of fact, bears only a very small ratio to) 



/.• 2kCt 



6 \_m + 2Gtkp (m + 2CtkpJ 



dp, 



which again is less than 



ikp 



k 2kCf 



-+ — — 

 m riv 



dp. 



m 



(49) 



Kemembering that the ratio of /*; Gtr to m is nearly unity, this is seen to be 



approximately equal to 



c^'- (1 + 2/At 



■m 



Accordingly, it ajDpears that the neglect of (32j in comparison with (31) 

 involves an error which, estimated as a fraction, is of order kj'in, and which 

 therefore is admissible. 



"We have still to consider the terms omitted from the first integral 

 in (29). 



The former of these, viz. : — 



[ {I,{kp)K^{kh) - I,{kh) K,(kp)] p-' sin m (p - h) sin /.; (z - Wt) dp (50) 



is less than 



Mkp)K,{kb)p-'dp; 



(51) 



and from what has gone before, it is evident that, since kr is large, this is 

 approximately equal to {2Trkh'^)~^ e^''Ki{kb) ; (52) 



and the neglect of this, in comparison with the product of ^^38) by k"^ + wr, 

 at times such as considered, involves a fractional error of order l/7??.Vl 

 The latter, viz. : — 



mkp)K,{kb) - I,{kb)K,{kp)] m cos m (p - b) sin k{z- Wt) dp (53) 



is less than 



mKi{kb) Ii{kp) dp, 



Jb 



(54) 

 (55) 



(56) 



i.e. than mK,(kb)k-'(f,{kr) - Io(kb)), 



which is approximately equal to (^iTrkh)"^ me^''Ki{kh) ; 



and the neglect of this involves a fractional error of order llmr. Thus, 



under the conditions stated, these terms may be neglected. The substitution 



of the product of (38) by k" + m' for the first integral in (29) has thus been 



justified. 



Again, in the multiplier of the first integral in (29), we may omit 

 Ii(kr)Ki(ka) in comparison with l^{ka)K^{kr), since e^C^"') is large, and on 



[7-1 



