455 
of Edinburgh , Session 1879 - 80 . 
coming nearer and nearer down to the 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 (which is equal to J' 0 ) from q = 0 to q=20* 
When ma 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 (g) = 0, which, according to Hansen’s 
Table, is 2*4049, or approximately enough for the present 2*4. In 
every case in which q is very large in comparison with ma, whether 
ma is small or not, (54) gives 
2 oimfl . , n 
n = approximately. 
Now going back to (6) we see that the summation of two solutions 
to constitute waves propagated along the length of the column, gives 
r = - g sin (nt — mz) ; rO — T + r cos (nt - mz) 
z — w cos (nt - mz) ; p = P + z? 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 (oa/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 
Koyal 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, 
- 3£ i (ma ) 
maUfma) 
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'.(g) _ i i 
qj t (q) q 2 4 
• ( 57 ). 
Repu Wished in Lommel’s “ Besselsche Functionen,” Leipzig, 1868. 
