58 



107. By substituting, in equation (21), the expressions for sin 8, and 

 €os 8 cos H, derived from equations (60) to (64), an expression for 

 cos d may be derived in terms of s, k, h, T, ^, and v. 



The astronomical formula for the correction k (in radians) is: 



k = 2e sin (.s— ;p) + 5/4 e^ sin 2(s— 2>) + 15/4 me sin {s — 2h+p) 



+ 11/8 m^ sin 2 {s-h) (65) 



and since k is small, its angle, in radians, may be substituted for its 

 sine. 



The astronomical formula for IjR, in equation (16) is 



l/i^=l/c + e cos (s-p)/c(l-e-)+e2 cos 2{s-p)lc{l-e') 

 + 15/8 me cos {s-2h^-p)lc{\-e'')+m^ cos 2{,s-h)lc{l-e'') (66) 



108. The expression for the lunar equilibrium tide in terms of the 

 angles X, T, s, h, ^, and v, and astronomical constants, is then derived 

 by substituting these expressions for cos 9 and l/R in equation (16) 

 and successively converting the products of the sines and cosines of 

 the angles T, s, h, ^, and v, into sines and cosines of their sums and 

 differences, by the application of the elementary trigonometric for- 

 mulas: 



cos X COS y—Yi cos {x—y)-\-% cos {x-\-7j) 

 sin X sin y=}i cos {x—y) — ^ cos (x+i/) 

 sin x cos y=y2 sin (j^+?/) + /2 sin (x—y) 

 cos X sin y=% sin ix-^y) — }^ sin (x— ?/) (67) 



cos^ a;=K(l+cos 2x) 



sin^ a;=K(l — cos 2x) 

 sin a; cos x=K sin 2a:; 



109. The result is an equation for u which contains 63 terms, and 

 which would cover more than a printed page. It is not here repeated. 

 But 21 of the variable terms have coefficients of sufficient numerical 

 value to require consideration. These give the following working 

 equation for the lunar equilibrium tide, now designated as y: 



?/=3/2 {Ma'lEc')aX 



{cos^ X cos* %! [Oi-5/4: e') cos {2T^2h-2s + 2^-2v) Mg 



+ 7/4 e cos (2r+2/i-3s+i? + 2^-2v) Na 



+ 1/4 e cos (2T+2h-s-p-{-2^-2p+180°) [U] 



+ 17/4 e^ cos- {2T+2h-4s+2p + 2^-2v) 2N 



+ 105/32 me cos {2T+4:h-3s-p+2^—2v) v-i 



+ 15/32 me cos (2r-s + 2?+2^-2;^+180°) X2 



