in an Annular Trough. 
203 
where A n is a constant different from the A n above, and with this 
form of 7] we can discuss the contour lines. 
The two simplest cases are the cases of n = 0 and n = 1. 
Case I. n — 0. 
7] = A 0 B 0 (/ cr ) cos mt. 
The motion is symmetrical about the origin, so that the waves 
have annular ridges and furrows, /c must satisfy (8 a) n=0 and when 
r is such that 
B 0 (/cr) = 0, 
there is a nodal circle. 
Case II. n — 1. 
ciri 
7) = A^B y ( /cr ) 6 . cos mt. 
v ' cos 
Besides the nodal circles given by 
B 1 (/cr) = 0, 
there is a nodal diameter 6 = 0 or 7t/2, whose position, however, is 
indeterminate since the origin of 6 is arbitrary. It does not 
follow that for every value of /c which satisfies (8a) n=1 , there will 
also be a nodal circle. 
Returning to the general case, the period is seen to be 
m V g/c 
and it will be noticed that T and the whole motion in general are 
independent of the density of the liquid. The form of the free 
surface along a line through the origin is given by 
7] = B n (/cr). 
Theoretically the problem is now completely solved, but for its 
numerical application and the tracing of the contour lines we must 
know the solution of equation (9) and have the means of evaluat- 
ing the Bessel’s functions of the second kind : for I have not been 
able to find tables of those functions. 
The solution of equation (9) will be found on page 242 of Gray 
and Mathews’ Bessel’s Functions and the expressions for F 0 and Y x 
which were used to calculate their values, are given on page 22 of 
the same book. 
F„ = log a; + 4 ji J,- i J t + | J, - ... j , 
r, . i. g . - i - J . + 4 j A j, - JL J, + J , . . ) . 
