Vibrations of a Columnar Vortex. 167 



The Subcase of i=l, and ma very small, is particularly 

 interesting and important. In it we have, by (42), for the 

 second member of (50), approximately, 



- *; (""0 = 4, fl + mV (log 1- + -1159)1 • • ( 56 ) 

 mavi{rna) ma L \ ma J A ' 



In this case the smallest root, q, is comparable with ma, and 

 all the others are large in comparison with ma. To find the 

 smallest, remark that when q is very small we have, to a 

 second approximation, 



^=4-1 (57) 



qj^q) q 2 4 v ; 



Hence (50), with i=l, becomes, to a first approximation, 



^(l+k) = -T* ( 58 > 



<f V X/ m z a z v J 



This and (52), used to find the two unknowns X and q 2 , give 



A = |-, and q 2 = 3m 2 a 2 , 

 for a first approximation. Now, with i = l, (51) becomes 



and therefore n/co is infinitely small. Hence (52) gives for 

 a second approximation, 



J»=8*W(1+|5), (59) 



and we have 



J-=-?4 ! fi-^) (60) 



q z X 3 roV V 3o)J k j 



Using now (57), (59), (60), and (56) in (50), we find, to a 

 second approximation, 



1 ■/ Bn\ 1.2/ 5n\ 

 3ma 2 \ 3co J 4 T Sma\ 2>J 



= -4-2 Tl + wVflog — + -1159)] , 



whence 



= ^v(lagi + i + -1159). . . (CI) 



Compare this result with (43) above. The fact that, as in (43)> 

 — n is positive in (61), shows that in this case also the direc- 

 tion in which the disturbance travels round the cylinder is 



