Orr — Stahiliti/ or Instahility of Motions of a Viscous Liquid. 119 



(,_ ru-jry. •\ . Exp{-Xa-vih) jExp (- \ct - u,) 



/v\¥„ f-v\H+3X2yi\ , ,,,,,^. .._, Exi^(ka + u^ ) 



\lpj \ o/p ) (-A + i((-^j -^pa^)/v)^) 



(A-t((-p+/j3ra]/i/)2 ^ \ -%{[-]) -lpa%) I v^ ) 



(72) 

 Wi denoting, as before, f (^/3/v)2 {- y(/j3) + «i}"5", and -2^2 the corresponding 

 expression with the sign of a changed. 



Aet. 27. Some Results. 



It appears that the period-equation has no roots for which the real part 

 of p (or even that of p') is positive. If the real part of p is supposed positive, 

 the equation assumes a simpler form ; the first expression within the [ } is 

 to be replaced by 

 / ///7/^^ -x 1 Exp{\a + u,) ^ ^ 1 Exp {- Xa + u^) 



X-\-i{{- 2/ + liiai)lv)^ \ 1 i\t-j / x + i[{-p'- l^ai) jvf ' 



(73) 

 and the third is to be similarly replaced by the first and last of the four 

 terms which constitute it. In fact, if the real part of p (though not neces- 

 sarily if merely that of p') is positive, that of any one of the expressions 

 ± Xy ± u either continually increases or continually decreases as y changes 

 from - a to + « ; and accordingly it seems evident that we may proceed as 

 indicated in the third paragraph of the preceding Article, and thus obtain this 

 modified form of the period-equation. If we now consider the terms in the 

 equation which are most important, it will be found that it is necessary that 

 Qi\a should be complex or less than unity, which is, of course, impossible. 



In using these approximate forms there is a tacit assumption that p is not 

 too near either of the values ± l\^ai : making the contrary supposition in this 

 case, too, I failed to obtain any evidence of the existence of a root whose 

 real part is positive. 



It may be shown that, if with the origin as centre, a circle be described 

 for which 



mod. 2a [-p'lvf = (2r + 1) 7r/2, 



where r is a large enough integer, the number of roots of the period-equation 

 for which ^:>' lies within this circle is r - 1.* This follows as in Art. 18 : the 

 alterations in the form of the left-hand member of (72) which have to be 

 made in different portions of the contour are, as in that Article, negligible if 

 p' is sufficiently great. 



* This is one less thau if tlie boundary -conditions included V'W = 0. (See Art. 18.) 



[16-^=1 



