64 Proceedings of the Boyal Irish Academy. 



decreases, but is simply carried round by the fluid, the velocity of each 

 element remaining invariable. 



Aet. 25. Disturbance having Stream-Function initially sin c^(r ^- & ^) sinsO; 

 for suitahle Constants it increases greatly. 



And as in the steady motions discussed in the previous chapters, we may 

 show in the case of some disturbances of analytically simple types that the 

 disturbance before dying out will increase, and increase very much. 



Consider a disturbed motion in which initially 



^ = ^0 = sin c- {r~- - h'"^) sin sB where sin c^ {cr- - h~^-] = 0. 

 Here 

 pfXp) +f:p) - s'p-'f{p) = 4.c-p-' cos r {p-' - b--) - (4c*^- + s>->) sin c' [p'' - b'^) ; 



(10) 

 and accordingly we have at time t 



2s {a'b-" - a-'b') if, = (-/-^a"* - r-'a') 



(p'b-'-p-'b') [4:C-p-'coscXp--~b-')-i4:c'p-'+sy-')&ii-ie-(p-^-b-') ] sins { 0-(C'+ C'p-')t } dp 

 +(r'b'' - r-'b') 



(p'a-'-p-h(') [ 4:C'p-'cosc-(p-'-b-')-[4:c'p-' + s-p-^)sinr(/3---&--) } sins ( 0-(C'+ C'p-')t ] dp. 



(11) 

 We will obtain the approximate value of x^ as given by this equation at 



the time when c^ = sC't. Consider the first integral in the right-hand 



member ; it may be expressed as the difference of two, thus : 



■■»• 

 i {p'b-'-p-'b') 



J b 



X j 4c'p-^sm[sd-sCt-c-b-'+{c'-sC't)p-^}+(4c'p-'+s'p'')cos{s9'sCt-c-b-H{c^^^ ]dp 



(p'b-'-p-'b') 



x\4:C-p-^sin{'c-^sC't)p-HsCt-s9-rb-^}+{4:c'p-'+s'p-')cos[{c'+sC't)p-^+sCt-se-c'b-'\ \dp. 



(12) 

 At the time referred to, the former of these, or 



i sin (sO - Cc-jC- c'b'') {' 4:c'p-'(p'b-' - p-'b') dp 



+ 1 cos (sd-CcyC'-c'-b-') 

 is equal to 

 2c^ sin [sB-CcyC'-c-b-'] 



(4cV + s-p-') (p'b-' - p-'b') dp, (13) 



b 



+ ^cos[sd CcyC'-c-b''\ 



r'-~b-' o'-'-'b' 2sb-^' 

 s- 2 s+ 2 s"-4_ 



4c* — r + ; -. — ^.l + s(')''b-'+ r-'b' - 2) . 



Is -4 s + 4 s'-lbj ^j 



(14> 



