Oscillations of Rotating Water. 115 



we please of the fundamental modes, and working out the whole 

 motion of the system for each. The roots of this equation, 

 which are found to be all real by the Fourier- Sturm-Liouville 

 theory, are the speeds* of the successiye fundamental modes, 

 corresponding to the different circular nodal subdivisions of 

 the i diametral divisions implied by the assumed value of i. 

 Thus, by giving to i the successive values 0, 1, 2, 3, &c, and 

 solving the transcendental equation so found for each, we find 

 all the fundamental modes of vibration of the mass of matter 

 in the supposed circumstances. 



If there is no central island, the solution of (19) which must 

 be taken is that for which P and its differential coefficients 

 are all finite when ?' = 0. Hence P is what is called a Bessel's 

 function of the first kind and of order i, and, according to the 

 established notation f, we have 



MVt) < 21 > 



The solution found above for an endless circular canal is 

 fallen upon by giving a very great value to t. Thus, if we 



put-^-=A, so that \ may denote wave-length, we have 



l 2*77" 



- = — -, which will now be the m of former notation. We 



r X ldh 



must now neglect the term --=- in (19); and thus the differ- 

 ential equation becomes r c r 



l 2 h=0, (22) 



(Ph . /**-4,cd 2 

 ch 

 or 



dr 2 



where P denotes m 1 ^ — . A solution of this equation is 



h=.c" ly , where y = a — r\ and using this in (20) above, we find 



* In the last two or three tidal reports of the British Association the 

 word " speed," in reference to a simple harmonic function, has been used 

 to designate the angular Telocity of a body moving in a circle in the same 



a 



period. Thus, if T be the period, -^ is the speed j vice versa, if a- be the 



speed, — is the period. 



cr 



t Neumann, Theorie der BesseVschen Functionen (Leipzig ; 1867), § 5 ; 

 and Lommel, Studien iiber die BesseVschen Functionen (Leipzig, 1808), 

 §29. 



