{(p%-^-p-%^)p'l 



66 Proceedings of the Boyal Irish Academy. 



be proved that the most important part of this, at the critical time when 

 c^ - sC't vanishes, is 



r '" 

 - i ip'b-' - p-'b') 4cy= cos (2cy- + c'Cl'G - c'b''- - s9) dp, (19) 



J b 



or, integrating hj parts, 



|g2 (^^.sj^-s _ .^.-sjjs-. ,^,-2 gij^ (2c-r- + c'C/C - c'h-' - sd) 



- ic' I sin {2c' p-' + c'C/C - c'Jr' - sd) -^ ! ip'b-' - p-%') p-- ; dp . (20) 



J i -P 



It is readily seen that the contribution of the first term in this to the right- 

 hand member of (11) is cancelled by a similar expression of opposite sign 

 which arises when the second term of (11 is treated in a similar fashion. 

 The second term in (20) is less — and, as a matter of fact, much less — than if 

 the sine were replaced by unity and 



d_ 

 Tp 



by its numerical value : and as 



{p'b-' - p-'h') p-\ 



continually increases from zero as p increases from b, it would then be 

 equal to - ^c'- {•,%-' - r-'b') -r\ (21) 



and the ratio of this to (18) is, unless the cosine in (18) be a very small 

 fraction, of order sr-c'-, which we have supposed small. 



In a similar manner it may be proved that the omissions which have 

 been made from the second term of the right-hand member of (11) in 

 replacing that member by (17) are legitimate. 



Making this substitution. (11) is thus sensibly equivalent to 



-^ i: - 2c^/-*.r- cos {sB - c'C/C - c'b-% (22) 



which exceeds its initial value in a ratio of order c*r-V-, which has been 

 supposed large. 



At the critical time the radial velocity 



u = d4lrdB = 2c^r-^s sin {s9 - c'C\0' - c'V) 



exceeds its initial value in a ratio of the same order, c^;'-Vl 



But, as in the problems of the preceding chapters, the velocity in the 



direction of flow cannot, at least -prima facie, be obtained by differentiating 



(22), for the reason that in this equation there has been neglected a term 



invohdng the angle 



2c-r-= 4- &CIC' - C-&-- - .50, 



and differentiation with respect to r introduces a relatively large multiplier. 

 By differentiating equation (II) it may be shown that at the critical time 



