462 Mr. Lubbock on Bernoulli's Theory of the Tides. 



Mm mR* n ' 2y . mR 3 \ a J 1 



= -^+7 27^ (1 ~ 3COS ?)_ 2r4 ° S? " 5COS? J 



+ — — 



m'R* 



2r 



m 



(l-Scos^O-^jscos^-Scos^j 



If a denote right ascension, & declination, <p geographical 

 latitude, and a sidereal time, 



cos £ = cos & cos f cos fa —a) + sin 8 sin <p. 



9 v i, • «.u cos8 8 cos 2 A 

 cos 3 ? contains the term 21 cos (2 a— 2 a). 



Hence, neglecting constant terms and those of the argument 

 a— a, &c, the height of high water 



3 m R 3 

 = D + jjg^S cos 2 8 cos 2 <p cos (2 a— 2a) 



3t»' i£ 3 



+ TMW cos2 * co&2 * cos (2 *~ 2a ') 



= D + £{^cos(2/x-2a) + cos (2 a -2 a')}, 



where D is a constant depending only on the zero line from 

 which the heights are reckoned. 



. m cos 2 a P 3 ^ „ , , M _,. 



^ = — . — -ra-sr* -E = GWcos 2 3'P' 3 , 



7W' COS 2 6' /* 3 



P being the horizontal parallax, and C a constant depend- 

 ing upon geographical latitude. 



By differentiating the expression for the height, in order to 

 find when the height is a maximum, the following well-known 

 formula is obtained : 



tan (2 a — 2 a') = _ • A — v la , — ~-x. 

 v r ' l+A cos (2a —2 a) 



The readiest method of calculating tables of the inequalities 



of the heights and intervals from the above, which coincide 



with Bernoulli's expressions, is to obtain the angle \|/ = a— a', 



from the expression 



A j A sin 2 $ 



tan 2vp = r— i T,-, 



l + Acos24> 



for given values of $, then the height of high water- 

 is Z) + jE {cos2\J/ + ^cos(2<p — 2^)}. 



The value which I formerly deduced from the London ob- 

 servations for the constant A with parallax 57', and when 

 a as 3', (see Phil. Trans. 1831, p. 387,) is -3788; log. 



