ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 171 



and the results superposed, so that the deformation at time t resulting from this dis- 

 turbing force will be given by 



* 8 =oo ,>=r , ia<+*^> , *, -;(*+$) r -a* v* -in 



r ? ? v y " h e i n "-s p s i p i ft -+ 2 P* i 



4 = 2~17 ** * * TT T, , ir. V + ~. fn-l T r,, + - r,, ,., + ... S . 



4< z ffl- 8 = = . L "i-I ~ L + K +2 I - J J 



The corresponding formulae when the depth is variable may be obtained by replacing 

 h, yl, 8;, H?,, K;, 14, a, 2/L by ir, G' rt , AJ, j&, it;, ' , tf, respectively, where 

 *r, G' n , &' n , f', >;' are defined by the equations (35), (41), (42), and 



fc 



J^., Cn 7 /!! g-2Vn-'J 



<*?" TTf 'W 



JLfl-2 ' 



m. _ . -^"-2 .l^L 

 n - . Tirt 



*n *+:> 



while A^ is obtained by writing S" n in place of y* in the right-hand member of (42). 



13. Classification of Tide*. 



In the last section we have reduced the problem of the evaluation of the forced 

 tides due to any disturbing force to that of the development of the disturbing poten- 

 tial as a series of surface-harmonics. This development for the case of the disturbance 

 of the ocean due to the attraction of the sun and moon has been already dealt with, 

 and reference may be made to Professor DARWIN'S article in the ' Encyclopaedia 

 Britannica' for a full account of it. We give here a short summary of the principal 

 results of which we propose to make use. 



The principal part of the disturbing potential will consist of spherical harmonic 

 functions of the second order, and when expressed by means of zonal and tesseral 

 harmonics the terms which occur will be of three tvpes characterized bv the rank of 



ti i. > 



the harmonic involved. 



For the first type s = 0, and the corresponding tides will be expressible as series of 

 zonal harmonics. For these types the value of X will be small in comparison with 

 that of CD, so that the period of the disturbance is long compared with a sidereal day. 

 The terms will cease to be oscillatory when the orbital motion of the disturbing body 

 is neglected. The tides generated by these parts of the disturbing potential have 

 been already dealt with in Part I. 



The terms of the second type are those for which s = 1 ; in certain of these terms 

 the value of X will be equal to w, so that the period is rigorously equal to a sidereal 

 day, while in the rest the period will reduce to a sidereal day when the orbital motion 

 of the disturbing body is neglected. If n denote the mean orbital motion of the 

 luminary, the " speeds" of the principal diurnal tides will be <u and a> 2n. We 

 propose to neglect the sun's orbital motion, so that for each of the principal solar 



z 2 



