67 



In the first term of equation (78), for example: 



C=3/2(Ma7£'c3)a(l/2-5/4 e") cos^ X, and (/.(/)=cos^ ){!. 



The ampHtude, J, of the equilibrium component fluctuates slowly 

 between fixed limits with the changing values of I. A rigid analysis, 

 which need not be here repeated, shows that the mean value of J 

 during the circuit of the moon's node around the ecliptic, is 



Jo=C [<^(7)]oX[cos 'A,=CM (90) 



where [^(/)]o is the mean value of <^ (7), [cos ti]o is the mean value of 

 cos li, and M is the numerical value of their product. 

 From equations (89) and (90) 



Jo/J=M/0(/) (91) 



124. Mean value of amplitudes of the actual components. — As a basic 

 assumption, the fluctuation of the amplitude of a component of the 

 actual tide with the changing values of I is proportional to the con- 

 current fluctuation of the corresponding equilibrium component 

 (par. 102). If then R is the value of the amplitude of a component 

 of the actual tide as determined from a particular set of observations, 

 H the mean value of the amplitude, and J the amplitude of the 

 corresponding component of the equilibrium tide when I has the value 

 prevailing during the period of the observations: 



H/R=Jo/J=Mfct>{I) (92) 



The factor Ml<j>{r) is conventionally designated as F. Its recipro- 

 cal, <^(/)/Mis designated as/. Hence 



H=FR R=-JH. (93) 



125. Expressions for F. — A reference to equation (68) shows that 

 the expressions for (/) and for u in the terms representing the 

 various components are as follows: 



Components 4>(I) u 



Mo, Na, 2N, V2, X2, M2, cos* )il, 2^-2v 



Oi, Qi, 2Q, pi, sin I cos2 YJ, 2^-v 



00, sin/sin^K/, -(2^+p) 



Ji, sin 2 7, —V 



Mf, sin2 7, -2g 



Mm, 1-3/2 sin^ 7, 



The mean values of these functions of 7, and of the corresponding 

 expressions for cos u are found by deriving the expressions for 7, ^, 

 and V in terms of A^from the spherical triangles OUI and lUUi, figure 



