of Cylindrical Fluid Surfaces. 



179 



corresponding in a similar manner to (9). In the second 

 case we have 



0O*) = (7+logi#) T oM 



x< 



x* 



)j &i ~ $\J? ^2 — 2\ 4 2 . 6 2 ^ 3 ~~* ' * ' ' ^^ 

 x<\>' (x) = I (x) + (y 4- log J x) x I 1 (a?) 



^2 ™4 ™6 



~ "2 °i ~" 22~J ° 2 "" 2 2 . 4 2 . 6 ^ ' 



-r. 



U*)- 



xiM. 



(10). 



•oooo 



-0(*). 



X<p'(x). 



(11). 



00 



1-0000 



-0000 



00 



1-0000 



•oooo 



01 



10025 



•0050 



•0703 



2-4270 



•9854 



•6339 



02 



1-0100 



•0201 



•1382 



1-7527 



•9551 



•7233 



03 



1-0226 



•0455 



•2012 



1-3724 



•9169 



•7795 



04 



„ 10404 



•0816 



•2567 



11146 



•8738 



•8113 



05 



1-0635 



•1289 



•30J5 



•9244 



•8283 



•8198 



06 



1-0920 



•1882 



•3321 



•7774 ' 



•7817 



•8022 



0-7 



1-1264 



•2603 



•3433 



•6607 



•7353 



•7535 



0-8 



1-1665 



•3463 



•3269 



•5654 



•6894 



•6625 



0-9 



1-2130 



•4474 



•2647 



•4869 



•6449 



•5017 



10 







•oooo 







•oooo 



On account of the factor (1— x 2 ) both (10) and (11) vanish 

 when # = and when a? =1. Beyond x = l, (10), (11) become 

 imaginary, indicating stability. It will be seen that when the 

 fluid is internal the instability is a maximum between x = '6 

 and x=-.'7 ; and when the fluid is external, between x = '± and 

 x = '5. That the maximum instability would correspond to a 

 longer wave-length in the case of the external fluid might 

 have been expected, in view of the greater room available for 

 the flow. The same consideration also explains the higher 

 maximum attained by (1.1) than by (10). 



In order the better to study the region of the maximum, 

 the following additional values have been calculated by the 

 usual bisection formula 



2 + 16 



X. 



(10). 



X. 



(11). 



•65 

 •70 

 •75 



•3406 

 •3433 

 •3397 



•45 



•50 

 •55 



•8186 

 •8198 

 •8147 



N2 



