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 = and n = \. 



Case I. n = 0. 



rj = A B (/cr) cos int. 



The motion is symmetrical about the origin, so that the waves 

 have annular ridges and furrows, k must satisfy (8a) n=0 and when 

 r is such that 



B ( K r) = 0, 

 there is a nodal circle. 



Case II. n = 1. 



sin 

 7) = AiB, («r) . cos mt. 

 cos 



Besides the nodal circles given by 



B, O) = 0, 



there is a nodal diameter 6 = or 7r/2, whose position, however, is 

 indeterminate since the origin of 6 is arbitrary. It does not 

 follow that for every value of k which satisfies (8a) n=1 , there will 

 also be a nodal circle. 



Returning to the general case, the period is seen to be 



m 2-7T /coth k/i 



1 — — = 'Lit 



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 (AT). 



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 and Fj 

 which were used to calculate their values, are given on page 22 of 

 the same book. 



Y Q = Jo log a? + 4 L J" 2 - | J" 4 + g J 6 - ... I , 



