172 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



diurnal constituents we shall suppose that X = ta rigorously. The same analysis will 

 then apply to one of the lunar diurnal constituents, while in order to illustrate the 

 effect of the departure of the period from exact coincidence with a sidereal day we 

 shall evaluate independently the lunar diurnal constituent for which 



X(= (a 2n) = 0-92700w. 



The principal part of the tidal oscillations will be due to the third part of the 

 disturbing potential, which involves harmonics of rank 2. The period for the tides 

 due to these terms will differ but slightly from half a sidereal day, and will reduce to 

 half a day exactly when the orbital motion of the disturbing body is neglected. We 

 shall therefore assume that X = 2<u rigorously for the solar semi-diurnal tides, while 

 we shall take the value 



X/2( = 1 w/ = 0-96350 



as typical of the lunar semi-diurnal constituents. The analysis applied to the solar 

 constituents will be rigorously applicable to the sidereal luni-solar semi-diurnal tide 

 usually denoted by the symbol rL. 



$ 14. Special Cases. 



Instead of making use of the equation (31) as we have done in 12, we may of 

 course compute the forced tides by means of the equations (49), (50) of 6, 

 determining incidentally the constants D'. Thus, if we suppose that all the y's 

 are zero except y" +1 , we have the following equations for the determination of Dj, 



C]' TV .trp 



**-/c 4. 1 JL-* v 4. *> LVU* 



I 2s + 5 v 



"-A ps |<r TV I v 9 + 2) 2 (2s + 3) f ^ t 



o ^.+i i> 8+2^8+2 T 9 S + 7 ' +3 



where the terms on the right are all zero, except in the second equation. Now it is 

 evident that if N* = 0, or 



2<M 



^ r 



all these equations will be satisfied if 



and 



