HARMONIC AISTALYSIS AND PREDICTION OF TIDES. 49 



Mean value of variable factors of coefficient 



= [cos* i/{(l-6 tan^ |/ cos 2P) cos {2$-2p) 



+ 6 tan^ 1/ sin 2P sin (2|-2j')]o (182) 



==tcos* il cos (2e-2j')-6 sin^ ^7 cos^ ^I cos (2P + 2f-2j/)]o 



Substituting the equivalent of P from (175), the last term of (182) 



3/2 sin^ / cos (2,p - 2v) (183) 



Now p increases uniformly throughout the entire circumference, 

 while / and v are functions of N, the period of which is incommen- 

 surate with that of p. It is evident, therefore, that in a series of 

 infinite length, the sum of the positive values of (183) will equal the 

 sum of the negative values, and the mean value of the term becomes 

 zero. The mean value of the first term is given by formula (155). 



For the mean value of the variable coefficient of the composite Lj 

 tide, we may now write 



r^^^cos {2$-2p-R)l=[cos*iIcos (2e-2i')]o 

 = cos* Jco cos* ^i = 0.9154 

 For the factor of reduction, 



(184) 



^ , cos^^g; cos* ^i _ 0.9154 . . 



F^^^^ = cos* ^7 -^^^V^^" ^ ^^ 



A comparison of (185) with (165) shows that 



7^^ of L2 = (7^ of M2) X 72a (186) 



14. THE Ml TIDE. 



In equation (100) we also have the terms {A)^^ and {A)^^ with a 

 difference in speed equal to twice the rate of change in p. 



Neglecting for the present the general coefficient applying to both 

 terms, we have 



term {A) ^g = 1/4 e sin 7 cos^ ^I cos iT+Ji-s-p + 2^-v- 7r/2) (187) 



term (A) ,, = 3/8 e sin 2l cos (T+h-s + p-v- 7r/2) (188) 



A reference to (99) indicates that the coefficient of the term (.4)28 

 is only about one-third that of term (A) 44. The latter will therefore 

 predominate and determine the mean period of the composite tide 

 formed by the combination of the two, while the former will intro- 

 duce certain inequalities in the resultant amplitudes and epochs. 



hetd=T+'h-s + p-Trl2-v (189: 



and P = 2? — 6 as in (175) 



