187] BESSEL'S FUNCTIONS. 307 



It appears from (i) that J s (z) belongs to the class of 'fluctuating 

 functions,' viz. as z increases the value of the function oscillates on both 

 sides of zero with a continually diminishing amplitude. The period of 

 the oscillations is ultimately 2ir. 



The general march of the functions is illustrated to some extent by the 

 curves y=J^(x\ y = J l (x}^ which are figured on the opposite page. For the 

 sake of clearness, the scale of the ordinates has been taken five times as great 

 as that of the abscissse. 



In the case s = 0, the motion is symmetrical about the origin, 

 so that the waves have annular ridges and furrows. The lowest 

 roots of 



J '(ka) = ........................... (7) 



are given by 



&/* = T2197, 2-2330, 3*2383, ............ (8)*, 



these values tending ultimately to the form ka/7r = m + %, where 

 m is integral. In the mth mode of the symmetrical class there 

 are m nodal circles whose radii are given by =0 or 



J (kr) = ........................... (9). 



The roots of this are 



AT/IT ='7655, 17571, 27546, ............ (10)f. 



For example, in the first symmetrical mode there is one nodal 

 circle r = '628a. The form of the section of the free surface by 

 a plane through the axis of z, in any of these modes, will be 

 understood from the drawing of the curve y J (x). 



When s > there are s equidistant nodal diameters, in addition 

 to the nodal circles 



J,(kr) = Q ........................... (11). 



It is to be noticed that, owing to the equality of the frequencies of 

 the two modes represented by (5), the normal modes are now to a 

 certain extent indeterminate ; viz. in place of cos s6 or sin sd we 

 might substitute coss(# s ), where a s is arbitrary. The nodal 

 diameters are then given by 



tf-*.-*^- 1 ,- ..................... (12), 



* Stokes, " On the Numerical Calculation of a class of Definite Integrals and 

 Infinite Series," Camb. Trans, i. ix. (1850), Math, and Phys. Papers, t. ii. p. 355. 



It is to be noticed that kajir is equal to T O /T, where r is the actual period, and T O 

 is the time a progressive wave would take to travel with the velocity (gh)% over a 

 space equal to the diameter 2a. 



t Stokes, I c. 



202 



