Oer — Stability or Instahility of Motions of a Perfect Liquid. 65 



The second integral in (11) may also be expressed as the difference of two, 

 of which one differs from (14) only in having h replaced by a, and in having 

 the opposite sign. 



When these terms * one from each of the integrals in (11), are combined 

 as in (11), the resulting contribution to the right-hand member of that 

 equation may be written in the form 



2s 



2c' 



s^-4 



2c* 



16 





sin(s0- CcyC'-c'b- 



cos[se-Cc-'/0'-c'h- 



15) 



1 1 



Not much information can be obtained from this without making some 

 definite supposition as to the relative values of a, b, r. If, for instance, 

 we suppose that b/a is nearly unity, it is evidently to be anticipated that the 

 results obtainable can differ but little from those which hold in the case of the 

 chief problem discussed in Chapter I. 



As another case in which results may be expressed with sensible accuracy 

 in a comparatively simple form take that in which hyr\ r^/a^ are both small. 

 The determination of the most important terms in the determinants in (15) 

 depends further on the value of s. If we suppose s large, the most important 

 terms in each are in order 



K - a'b-\ 



- cc 



- ct" 



so that the approximate value of (15) under these circumstances is 



- 2sa'b-' 



|^^sin(s0-r6y6"-c=^;-^)+('-|^ + l]co^{sd-c'CIC'-c'b-') 



(16) 

 and if we now further suppose cV"V^ large, this is sensibly equal to 



- 4.a'b-'r-'c's-^ cos [sB - c'CjC - c'b~^) ', (17) 



this result is equivalent to the replacing of (14) by the solitary term 

 2c^T'-^b'' 



Qo&{sB-c'ClG'-c'b-'), 



(18) 



and the treating of the second integral in (11) similarly. 



It may next be shown that, with the conditions stated, the other terms 

 are negligible, which would be omitted in thus replacing the right-hand 

 member of (11) by (16). In the integral which occurs in the first term in 

 this right-hand member we have omitted the second term of (12). It may 



* I.e. (14) and the analogue obtained from it by changing b into a, and changing sign. 

 R. I. A. PROC, VOL. XXVII., HECT. A. [9] 



