54 U. S. COAST AND GEODETIC SURVEY. 



Neglecting the eccentricity of the moon's orbit as being unim- 

 portant for this tide, we may take the mean distance c of the moon as 

 the equivalent of the actual distance d, and the mean longitude a 

 measured from the intersection as the equivalent of the actual longi- 

 tude I from the same origin. Substituting these equivalents and (98) 

 and (99) in (207) we obtain: 



3/2^-j5/12 cos^ X cos« i / cos (3r+3/i-3s + 3f-3j^) 



M3 (0.0107) 

 + 5/4 sin X cos^ X sin / cos* i / cos (2 T+ 2h - 3s + 3| - 2v + T/2) 



(0.0051) 

 + 5/4 sin X cos^ > sin /cos* \ /cos (2 r+ 2/i. - s + ^ - 2i^ - tt/j) 



(0.0051) 

 + {5/4 cos^ X cos« i /-cos X cos^ i /} cos {T+Ti- s-V^-v) 



[MJ (0.0100) 

 + {5/2 sin X cos^ X sin / cos* ^ /-sin X sin /} cos (s-^-tt/j)] 



(0.0116) (208) 



The maximum theoretical value in feet of the amplitude of each 

 term, when / has its mean value, is given after the term. For the first 

 term the maximum amplitude will apply to the Equator, where cos 

 X— 1; for the second and third terms to latitude X = cos~^ V 2/3, where 



sin X cos^ X will have the maximum value of 2/3Vl/3; for the fourth 



2 



term to latitude X = 003-^-7== rv^' '^^^ for ^^^ last term to lat- 



V15 cos^ t -' 

 itude 90°, where sin X=l. 



It will be noted that the first term containing 3 Tin its argument is a 

 terdiurnal component with a speed exactly three halves that of the 

 semidiurnal Mj term (^)i of (100). This component is usually 

 designated as M3. The fourth term is a diurnal component with a 

 speed exactly one half that of Mj. This component might be appro- 

 priately designated as Mj but a distinction should be made between 

 this and the composite Mj described in the preceding section. The 

 Ml depending upon the fourth power of the moon's parallax is usually 

 ignored in the analysis, as its eft'ects are negligible. All of the terms 

 of (208) are so small that they are of no practical importance in the 

 analysis and predictions of the tide. The component M3, however, 

 being obtained with very little additional labor when analyzing for 

 M2, is usually evaluated. 



The mean value of the variable coefficient of M3 is 



[cos« ^ /]o [cos (3^-3«^)]o (209) 



Developing in a manner similar to that described in section 11, 

 we find 



[cos« i /]o = cos« i CO [1 + 3/2 ^^ ^^^^J (210) 



[cos (3|-3.)]o= 1-9/4 ^^ -J-^|^ (211) 



[cos" i/]o [cos (3f-3v)]„ = cos« ico [1-3/4 i2] = cos» h " cos" i^ 



= 0.8758 (212) 



