36 THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



depend necessarily upon the time of the transmission of the tide from one place to 

 another, and therefore to increase as we follow the tide. But it appears that though 

 at Portsmouth this retardation is intermediate between that at Brest and at London, 

 as it might be expected to be from the course of the tide-hours, yet that at Plymouth, 

 where the tide is five or six hours earlier than it is at Portsmouth, the retardation 

 impHes a tide as late as London. 



Leaving, however, these anomalies to be removed or confirmed by the accumulation 

 and discussion of observations, I proceed to the effects of parallax and declination. 



1 . If from any cause U receive a small increment ^ Ji, we can easily find the corre- 

 sponding change, Han 2 {&' — X'), in the first side of equation (1.). We have 



V ^ / ^r fx h sin 2 (fl — a) . 8 h' 



I tan 2 {& — X') = 77; ^ ^/ 0:5 



^ ^ Qi' + A COS 2 (9 — a)) 



Let h' represent the mean value, and ^ h' any deviation from the mean ; and let S' 

 represent the semimenstrual inequality, that is, the value of 6' — X' freed from effects 

 of declination and parallax. Then 



« 0/ ^^ sin 2 (<p — a) 

 tan 2 !S = ,, , , ^^^ ,, 7- r ; whence 



n' + n cos 2 (<p — a) ' 



V. « /v n tan2 2S' Bh' 



Uan2(^--X-) = - 3.„,(^_^) ,-^ (2.) 



Now, coeteris paribus, h' is as the cube of the parallax. If, therefore, P be the mean 

 parallax, and p any other, 



A' + U' = ^'f' = /^' (1 + ^^)' = /i' + 3 A'^^, 



omitting other terms, because /> ~ P is small. 



Hence hh' =:3h' - — p — ; and when we make this substitution, equation (2.) gives 



the change in the first side due to the effect of lunar parallax. 



Since the arc 0' — X' is small, we may put it for its tangent ; hence, making the 

 above substitution and calling the effect of lunar parallax P', 



^^ — /? sin2(f-«)''^ P • 

 As a first approximation to the general form of the result, we may put -n sin 2 (^ — y) 



for tan 2 S', since tt is a fraction (about one third), and since the general course of the 



two functions, sin 2 (ip — a) and tan 2 S', agrees. 

 Hence we should have 



P' = — j^,~ sm 2 (^ — a) . ^ p 



P'= (P— j9).Bsin2 (9 — «); 

 B being a constant quantity. 



