1S90.] Harmonic Analysis of Tidal Observation*. 



will be multiplied by - VT = r~ an ^ * ne l &8 t term by the similar 

 ( + l) sin e 



function with z in place of e; also the r in the arguments must be 

 put equal to ^n. 



If the ( + 2) th tide falls exactly a semi-lunar period later than the 

 first, (n + l)e = TT. On account of the incommensurability of the 

 angular velocity <r, this condition cannot be rigorously satisfied, but 

 if the whole series of observations be broken up into such semi- 

 periods, then, on the average of many such summations, it may be 

 taken as true. 



Since \e is a small angle, ( + l) sin ^e = ^ir, and sin(n + l)e = 1 ; 

 hence the first factor is equal to 2/?r. 

 Again, 



= i(n+l)e . - = ^ . * ? = 76 32' in degrees; 

 6 a 



and (n 

 Therefore 



. r = sin 76 32' = X M436 = X X, suppose. 



1T1 ~~ 7T ffXrj 7T IT 



Again |ne = l7r-le = f7r-l -7048, 



friz = i_ 13-4647-l-4497 = ^7r-14-9144. 

 Now let 



= a- 1 -7048,] 



B = b- ] -7048, Y (23), 



7 = c-14-9144J 

 And we have 



+ |ne = 



(24). 



Thus, if n + 1 is the mean number of tides in a semi-lunar period, 

 the means of equations (22) become 



. ^ 



= R sin at R sinS \Rpsm 7, 



} (2--), 

 -R' cos et + Ro cos /3 XE/cos 7, 



where the summations are carried out over the first semi-lunar 

 period, which may be designated as 1. 



