Vibrations of a Columnar Vortex. 



163 



exceeding 6 or 7 the semiconvergent expressions* will give 

 the values of the functions nearly enough for most practical 

 purposes, with much less arithmetical labour. 

 From (37) and (39) we find, by differentiation, 



T , v mr , mrr- , m v , f 



F^mr): 



2 2 .4 ' 2 2 .4 2 .6 

 >m 4 r 



(40) 



2 + 2 3 .4 ' 2 2 .4 2 .6 



+ &c. 



> (41) 



^0^)= — -w[-l + 2(S X + -1159315)] 



+ P^[-1 + 2(S 2 + '1159315)] + &c. 



1 /mr wiV mV „ \ 

 a mr \ 2 2-. 4 2% 4"b / 



F 1 (mr)=^-i[-3 + 2(S 1 + -1159315)] 



+ ^J[7-6(S 2 --1159315)] + &o. 



For an illustration of Case II. with z = l, suppose ??ia to be 

 very small. Remarking that Si=l, we have 



ju / n 1+ ^ [log — -i + -1159] 



i^(ma) ~ m 2 a 2 n i rTTTTTTi 



1 --^L los ^ +i+ ' llo0 J 



= l + mV(logi i +4159). . (42) 



Hence in this case, at all events, N > i 2 ; and the angular velo- 

 city of the slow wave, in the reverse direction to that of the 



notation, to 23 places as follows : — 



1-90351 002G0 21423 47944 099. 



Thus it appears that the last figure in Stokes's result (10G) ought, as in 

 the text, to be instead of 2. In Callet's Tables we find 



log e 8 = 2-07944 15416 79835 92825 ; 



and subtracting the former number from this, we have the value of E to 

 20 places given the text. 

 * Stokes, ibid. 



N2 



