38 



THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



in considering the effect of solar declination, we should in like manner have k cos' / 

 for hy whence the value of tan 2 S' in equation (1.) would remain unaltered. 



Putting, as before, — j sin 2 (<p — a) for tan 2 S', the equation becomes 



k' 

 2 Q' = ^ cos Zsin 2 ((p — u) . s'm^ § ; or 



Q' = Dsin2 (9 — a).sin'§, 

 where D is a constant quantity. 



In this expression the correction for declination is supposed to be applied to the 

 time of high water, calculated for the moon and the place, both in the equator. But 

 in our tables this correction is applied to the mean place. Let A be the value of the 

 declination at this mean place ; then the correction for that case is D sin 2 (^— y) sin' A, 

 and therefore, 



Q' = (sin« ^ - sin' A) D sin 2 {(p — a) 

 is the correction to be applied to the mean. 

 The correction according to observation was 



Q' = (sin' § - sin' A) (C + D sin 2 (<p - y)). 



The second term agrees with the theory, except that the arc y is different from a : 

 the first term, C (sin'^ — sin' A), has nothing corresponding to it in the theory. 



3. We now proceed to the theoretical laws which regulate the height of high water. 



If 0, & be the distance of any place in the equator from the places to which the sun 

 and moon are vertical (these luminaries being supposed to be in the equinoctial), the 



height of the water at the place will be \ (h cos 2 (^ — X) + ^' cos 2 (p — V)) above 

 the mean level ; and if & be taken the distance of the highest water from the moon, 

 then 



h cos (2 ^ — X) + A' cos 2 (^ — X') 



will be the whole tide, which call y. 

 Now we have 



tan 2 r^' - X'^ - -^iHlKi^li^ 



where ^ -f- ^' = (p. 

 Hence we find 



cos 2 {p — X') 

 cos 2 (^ — X) = 



h' -\-h COS 2 ((p — a) 



V {A'2 + 2A;i'cos2 (^ - «) + A2} 



A-f A' cos 2 (^ — a) 



*/{W ^ 2A;i'cos2(f-«) +^^} 

 i/= V'{A'2^2A/i'cos2(^- a) + ^2} . (3.) 



If, as before, ^y represent the variation of 3/ in virtue of any variation of A', 



8y = '' + ''°;^'^--'> .U' (4.) 



y 



