Substituting in equation (13) 



A 1_ [Hq sin (xt - ho sin oc(t +£)] = 1.113 a ( ^g^ )^ . 



[sin cx (t +£) + 1/7 sin 3 o( (t +£) + 5/77 sin 5 <x (t +£) 



^ ] (U) 



Substituting t' for o( (t +£ ) and differentiating equation (14.), 



Ho cos (f -a£) - ho cos t' = a . 1.113 ( ^^^o) ^ f series] 



letting (2N)2 = a 1.113 (2g )^ 



A cX M 



Ho cos t' cos o^ £ + H sin t' <x£ = h^ cos t» = (2Nho)2 [seriesj (I5) 



When cos t' = 1, 



H^ cos oiS = ho (16) 



and when cos t' = 0, 



Hq sin<?c£= (2Nho)^ [ 1 - l/7 + 5/77 - + . . .] (17) 



The series converges to unity. Squaring and adding equations (16) and (17) 



Hq^ cos^o<£ = ho^ + Ho^ sin^o^e - 2NhQ = 



Ho2 - ho^ = 2Nho = 



ho = (n2 + Ho^)| - N (IS) 



Also 



2 1 

 cos c<e = (1 + N )2 - N 



H? Ho (19) 



The equations for the range in tha tide gage well are; 



H^'2 = Ho^ _ ho2 = Ho2 - [ (N^ -. Ho^) - 2N (N^ 4 H^2)i ^ ^] 



= 2N [ (N^ + Ho2)2 - n] = 2Nho (20) 



tan oc = - cot oc £ (21) 



The diraensionless equations are 2 



28 



