502 VISCOSITY. [CHAP, xi 



system on the elevated water is equivalent to a couple tending to 

 diminish the angular momentum of the system composed of the 

 earth and sea. 



In the present problem the amount of the couple can be easily 

 calculated. We find, from (9) and (11), for the integral tangential 

 force on the elevated water 



2ira 



pXrjdx = Trpahfsm 2v (12), 



o 



where h is the vertical amplitude. Since the positive direction of 

 X is eastwards, this shews that there is on the whole a balance of 

 westward force. If we multiply by a we get the amount of the 

 retarding couple, per unit breadth of the canal*. 



Another more obvious phenomenon, viz. the retardation of the time of 

 spring tides behind the days of new and full moon, can be illustrated on the 

 same principles. The composition of two simple-harmonic oscillations of 

 nearly equal speed gives 



T} = A COS (a-t + e) + A' COS (<r't + e') 



= (A + A' cos <) cos (a-t + e ) + A' sin </> sin (a-t + e) 



where d> = (V o-'X + t-f' .. 



If the first term in the second member of (i) represents the lunar, and the 

 second the solar tide, we shall have o-O', and A>A'. If we write 



A + A' cos = C cos a, A' sin = C sin a ............... (iii), 



we get r) = Ccos(<rt + f-a) .............................. (iv), 



where C=(A 2 +2AA'cos<f>+A'rf ........................ (v), 



A' sin d> 



and a=tan " 



This may be described as a simple-harmonic vibration of slowly varying 

 amplitude and phase. The amplitude ranges between the limits A + A', 

 whilst a lies always between \n. The 'speed 3 must also be regarded as 

 variable, viz. we find 



da = ovt 2 + (o- + <r') A A' cos + tr'A'* 



dt A 2 +2AA' cos 



* Cf. Delaunay, " Sur 1'existence d'une cause nouvelle ay ant une influence 

 sensible sur la valeur de 1'equation seculaire de la Lune," Comptes Rendus, t. Ixi. 

 (1865); Sir W. Thomson, "On the Observations and Calculations required to find 

 the Tidal Ketardation of the Earth's Eotation," Phil. Mag., June (supplement) 

 1866, Math, and Phys. Papers, t. iii., p. 337. The first direct numerical estimate of 

 tidal retardation appccirs to have been made by Ferrel, iu 1853. 



