455 
of Edinburgh, Session 1879 - 80 . 
coming nearer and nearer down to tlie roots of J 0 , the greater they 
are. They are easily calculated by aid of Hansen’s Tables of Bessel’s 
functions J 0 and J/ (which is equal to J' 0 ) from q = 0 to q = 20.* 
When met is a small fraction of unity, the second member of (53) is 
a large number, and even the smallest root exceeds by hut a small 
fraction the first root of J 0 (^) = 0, which, according to Hansen’s 
Table, is 24049, or approximately enough for the present 24. In 
every case in which q is very large in comparison with ma , whether 
met is small or not, (54) gives 
2 (oma . , , 
n = approximately. 
JSTow going back to (6) we see that the summation of two solutions 
to constitute waves propagated along the length of the column, gives 
r = - § sin (nt - mz ) ; rO = T + r cos (nt - mz) 
z — w cos (nt - mz) ; p = P + cos (nt - mz) 
The velocity of propagation of these waves is n/m. Hence when q 
is large in comparison with ma , the velocity of longitudinal waves 
is 2 ua/q, or 2 /q of the translational velocity of the surface of the 
core in its circular orbit. This is 1/1*2, or f of the translational 
velocity, in the case of ma small, and the mode corresponding to the 
smallest root of (53). A full examination of the internal motion of 
the core, as expressed by (55), (13), (48), (15) is most interesting and 
instructive. It must form a more developed communication to the 
Royal Society. 
The Sub-case of i= 1, and ma very small, is particularly interesting 
and important. In it we have, by (42), for the second member of 
(50), approximately, 
- 1 [(ma) 
malfma) 
1 + m 2 a 2 ( log 
ma 
+ *1159 
(56). 
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 approxi- 
mation, 
j ',(?) _ i _i 
(57). 
Republished in Lommel’s “ Besselsche Functionen,” Leipzig, 1868. 
