HARMONIC ANALYSIS AND PREDICTION OF TIDES. 55 



The factor F of Mg is therefore 



cos^ ^ CO cos' I i ^ 0.8758 . „v 



cos^ i / ~ cos® \ I 



Comparing (213) with (165) we find 



Fof M3 = [Fof Ma?/^ (214) 



16. SOLAR TIDES. 



The development of the term in (74) that represents the approxi- 

 mate solar tide will be similar to that for the lunar tide. By making 

 the proper substitutions of the solar elements for the lunar elements 

 in (10.0) a corresponding expression for the solar tide may be obtained. 



For mass of moon ( M) substitute mass of sun {S) . 



For mean distance of moon (c) substitute mean distance of sun (cj . 



For eccentricity of moon's orbit (e) substitute eccentricity of earth's 

 orbit (ej. 



For inclination of moon's orbit to Equator (/) substitute obliquity 

 of ecliptic {(d) . 



For mean longitude of moon (s) substitute mean longitude of sun (Ji) . 



For mean longitude of moon's perigee (p) substitute longitude of 

 sun's perigee (pj. 



For longitude of intersection of moon's orbit with Equator, in the 

 Equator and in the moon's orbit, v and f , respectively, substitute zero, 

 the longitude of vernal equinox. 



All terms in (100) representing the evection and variational ine- 

 qualities in the moon's motion may, of course, be omitted, and also 

 because of the small eccentricity of the solar orbit all elliptical terms 

 of the second power of e^ and elliptical terms of the first power of e^ 

 when combined as a factor with a sine function of the angle (v are 

 negligible. In terms such as {A)^ and {A)^^, where the second power 

 of e^ is a part of a larger coefficient, the entire coefficient is retained. 



With the above-named substitutions and omissions, the following 

 formula is obtained for the equilibrium height of the solar tide: 



y = 3/2^('^y acos^XX 



(5)i [(1/2-5/4 ei2) cos^ 1/2 w cos (2^) S2 (0. 3716)l 



(B)2 +7/4 ei CDS'* 1/2 w cos (2r-;i+pi) T2 (0. 0218) 



(B)3 +l/4e, cosM/2 CO cos (2T+/i-pi+7r) R2 (0. 0031) 



(5)i7 +(1/4+3/8 6^2) sin2 « cos i2T+2h) [Kj]^ (0. 0321) 



+3/2 I (^y a sin 2XX 



(B)26 [(l/2-5/4ei2) sin w cos^ 1/2 w cos {T-h + Trl2)__ Pi (0. 1542) 

 (B)34 +(1/2-5/4 ei2) sin w sin^ 1/2 wcos {T+3h-wl2)^ (0. 0066) 



(5)42 +(1/4+3/8 ei2) sin 2 co cos {T+h-7r/2) [KiY (0. 1478) 



+3/2 1 (^)' a (1/2-3/2 sin^ X) X 



(215) 



(B)5i [(1/2-5/4 ei2)sin2 CO cos (2/i) Ssa (0. 0640) 



(5)59 +(1/3 + 1/2 ei^) (1-3/2 sin^co) (0.2057) 



(5)60 +ei (1-3/2 sin2 co) cos (/i-pi)] (0.0103) 



The subscript of the notation at the left of each term refers to the 

 corresponding term in the development of the lunar tide. The nota- 

 tion at the right gives the usual designation of the component repre- 

 sented, the brackets indicating that the term only partially represents 



