ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 175 



15. Solar Semi-diurnal Tides. 



In the last section we have considered some special cases in which the tide-height 

 is expressible by a single term. In general it will however only be expressible by a 

 series of terms. It may be shown, as in 5 of Part I., that this series will be finite 

 when the law of depth is such that 



where n is an integer, n s being even or odd according as we are dealing with the 

 symmetrical or asymmetrical types of rank s. The values of I determined from this 

 equation will in general involve X, but for large values of n, they will approximate to 

 the same values of I as those required for the expression in finite terms of the lon<r- 

 period tides, since for such values 2s/X will be small compared with n (n +1). 



In other cases the expression for the tide-height will involve infinite series. We 

 deal in the present section with the case where I is zero, so that the depth is uniform. 



The numerical computation of the semi-diurnal tides admits of special simplicity 

 when X = 2w exactly. Putting X = 2w, s = 2, in the formula? (30), (29) we obtain 



i2 n 1 ., ii + 4 



~ (n + 3) (2n + 1) (2n + 3) ' = n (2n + 3) (2w + 5) ' 



. 8 (n - 1) (n + 2) ( - l)^ (n + 2) (n + 2) 2 (ft - 1) 



A,, = - 



ft 3 (n + I) 2 n- (n + 1) (2n - 1) ('In + 1) n (n + I) 2 (In + 1) ('2n + 3) ' 



the last of which gives on reduction 



A 2 , = 



,2 2 Qt- !)(* + 2) 



+ 3) ' 



This general formula for A'f, fails to hold when n = 2 ; for, in this case, n s and 

 n(n 1) 2cos/X are both zero, and therefore the second fraction involved in the 

 expression for A'f t is indeterminate. To determine its limiting form we must first 

 suppose the period slightly different from half a day, so that X is not rigorously equal 



to 2ca ; the formula (29) then gives 







*> _ 2 8 ~ 



= 



2 2 .3 3 3 s . 5.7(3.4 - 



which, on putting X = 2o>, reduces to 



X2 _i_ !- 43 JL 



" 2 3 .3 2 " 3 3 .7.10 5.7 ' 



The formulae for xl, yl, A* are now in a convenient form for logarithmic computa- 

 tion, and we may readily deduce the following numerical values 



* C/. LAPLACE, 'M^c. Gel.,' Part 1., Book IV., 7. 



