HAEMOIil'IC ANALYSIS AND PREDICTIOlSr OF TIDES. 41 



argument cause the intervals between successive maxima and minima 

 to vary slightly in length. In the analysis of a succession of short 

 series the amplitude obtained from each has the maximum value of 

 the function which occurs when (V+u) is or a multiple of 27r;. 

 but in the analysis of a single long series covering a great many 

 years the resulting amplitude represents the average of the values of 

 the function when V equals or a multiple of 27t. It will be readily 

 seen that the value of (102) is J when (V+u) =0 or a multiple of 

 27T, and J cos u when F=0 or a multiple of 27r. 



The expression "mean value of coefficient," as applied to the terms 

 in the formula for the equilibrium height of the lunar tide, is usually 

 taken to represent the mean value of the product / cos u. With the 

 value of u small, cos u has a value near unity and the mean value of 

 J cos u differs but little from the mean value of /. In the practical 

 application of the equilibrium theory to the harmonic analysis and 

 prediction of the tides it is of no consequence whether the mean 

 value J alone or of the product J cos u be taken as the mean coefficient,- 

 but for the sake of uniformity in representing the results the practice 

 heretofore adopted will be followed. 



With the factor cos u always near unity, the mean value of the- 

 product J cos u can be shown to be approximately equivalent to the 

 product of the mean value of each, and is so taken in the computa- 

 tions that follow. 



Referring to (100), the mean value of the following variables will 

 be required: 



cos* i I cos (2^ - 2u) for terms (A) ^ to (A) ^ 



sin^ 1 cos 2u for terms (A)^j to (^)i9 



sin / cos^ i I cos (2f — v) for terms (^)26 to {A)^^ 



sin / sin^ ^ I cos {2$ + v) for term {A)^^ 



sin 2 / cos V for terms (A) ^^ to (A) ^g 



sin^ I cos 2$ for terms {A)^^ to {A)^,^ 



(1—3/2 sin^ /) for terms (A) 59 to {A)^^ 



The first step will be to express the functions of /, v, and ^ in terms, 

 of N, the longitude of the moon's node. The latter changes uni- 

 formly with time, and it is in reference to time that the mean values, 

 are desired. 



Referring to Figure 6, the following formulas may be readily 

 derived from the spherical triangle QtA. Noting that the side 

 ^ T = iV, the longitude of the moon's node ; side ^A= Q,t' — ^== ^^f 

 — ^=N—^; and side '^A = v; and that the opposite angles are {tc — I)^ 

 0), and i, respectively; we have 



cos /= cos i cos w — sin -i sin w cos N (103) 



tani[(iV-f)+.]=^5|^taniiV (104) 



tanHW-f)-.] = i|{^tani;V (105) 



tan. . . ^V'°^ ^ (106) 



COS I sm (I) + sm i cos co cos iV 



tan {N-0 =—. ^-^^ ^, (107> 



cot w sm t + cos I cos N ^ '^ 



