HABMOlSriC AIJ'AL.YSIS AND PEJEDICTIOlSr OF TIDES. 31 



Substituting (82) and (83) in (76) , we have 



y = 3/2^ j[^ cos2 X cos* ^ I cos (21 - 2x) 



+ i cos2 X sin* i / cos {2l + 2x) 



+ 1 cos^ X sin^ / cos 2% 



+ ^ sin 2X sin / cos^ ^ / cos (2Z — % — 7r/2) 



+ 1 sin 2X sin 7 sin^ ^ I cos (2Z + x — 7r/2) 



+ J sin 2X sin 2 7 cos (x - 7r/2) 



+ (i- 3/2 sin^ X) (i sin' 7 cos 2Z) 



+ (i-3/2 sin^ X) (1/3-i sin^ 7)1 



(84) 



In the above formula d, the moon's actual distance from the earth 

 and I, the moon's true longitude in its orbit measured from the inter- 

 section, although functions of time, do not vary uniformly because 

 of certain inequalities in the motion of the moon. It is desired, 

 therefore, to find expressions for these quantities in terms of elements 

 that will vary uniformly with time. 



Letting a = mean longitude of moon in radians measured from the 

 intersection, then I being the true longitude from the same origin, we 

 may obtain from Table 1 



Z = (r + 2e sin (s-^) +5/46^ sin 2(s-p) 

 + 15/4 me sin is-2h + p) 

 + 11/8 TO2sin2(s-^) (85) 



Letting c=mean distance of moon from earth we may also obtain 

 from Table 1 



^=~ + a' e cos (s — p) + a' e^ cos 2 (s — p) 



+ 15/8 a' me cos {s — 2h + p) 



+ a' m^ cos 2 (s-h) (86) 



in which a' = — t^ sr • 



c (1 — e^) 



A reference to Table 2 will show that the quantities e and m may 

 each be considered as fractions of the first order and the product of 

 the two or the square of either as fractions of the second order. In 

 the following development terms smaller than those of the second 

 order will be neglected. 



Substituting the value of a' in (86) and multiplying by c, we obtain 

 after neglecting terms containing powers of e above the second 



-T = 1 + e cos {s — p)+ e^ cos 2 (s — p) 



+ 15/8 me cos (s — 2'h + p) 



+ m" cos 2 (s-h) (87) 



Cubing (87), 



-T3 = 1 + 3 e cos (s — p) +3 e^ cos' (s — p) 



+ 3 e' cos 2 (s — p) + 45/8 me cos (s — 2h^ p) 

 + 3 m' cos 2 (s-h) 



= 1 + 3/2 e^ + 3e cos (s-p) +9/2 e" cos 2 (s-p) 



+ 45/8 me cos (s-2'h + p) +Z m" cos 2(s-ti) (88) 



