Theory of the Tides 281 



forced by the tide-producing forces in an ocean covering the entire earth. 

 According to the selection of s, there are various kinds of these waves. 



(1)5=0 gives, according to the designation of Laplace, the "oscillation 

 of the first species , \ These oscillations are independent of the rotary motion 

 of the earth and depends wholly upon the motion of the disturbing body in 

 its orbit. The periods of these oscillations are very long and to these belong 

 particularly the fortnightly lunar tide and the semi-annual solar tide. 



(2) 5 = 1 gives the "oscillations of the second species", the most important 

 of which is the diurnal lunar and solar tides. 



(3) 5 = 2 are the "oscillations of the third species": which include the 

 semi-diurnal lunar and solar tides. 



We can only explain here the fundamental solutions for these three types 

 of oscillations, following the simplified explanations of Kelvin (1875, p. 279); 

 Airy (1845); Darwin (1886, p. 377); Lamb (1932, p. 330). 



If we put 



, a oj 2 R 



n-r\=n , --=/, — = m , 



(IX. 7) 

 a 4mR A co 2 R 2 



p = —, — = 4 —j— and a = coso 

 h gh 



a simple computation in the case 5 = gives as the differential equation for r\ 



£(£$S9 + *-°- (IX - 8) 



If, in this equation, we assume r\' = r\ we obtain the free oscillations of 

 the ocean. If there were no rotation (/ = oo and j3f 2 = a 2 R 2 /gh), the free 

 oscillations would be given by a 2 = n{n-\-\)ghlR 2 (n an integer); the simplest 

 oscillations have the form of spherical surface harmonic functions. In the 

 case of rotation, Laplace introduces in (IX. 8) a solution of rj in series of 

 increasing powers of fi and determines their coefficients from the boundary 

 conditions. This leads to their determination in the form of infinite continued 

 fraction. This famous solution of Laplace has led to controversies between 

 Airy (1842), Ferrel (1874) and Kelvin (1875, p. 227); however, the latter 

 has given the proof of its correctness and has developed it. 



Under natural conditions, these continued fractions, and the series for r), 

 converge rapidly. For the shortest period of this oscillation, for which the 

 elevation of the free surface behaves like a zonal spherical harmonic function 

 of 2nd order P 2 (cos&) we obtain, e.g. for a depth of 17,700 m and 2210 m 

 the values 1 1 h 35 min and 32 h 49 min respectively. 



The forced oscillations of the first species (s = 0) start with the potential 



rj = A x {\ — cos 2 #)cos(ot-M) . 

 The determination of the coefficients to solve r\ is made in a similar way as 



