1890.] Harmonic Analysis of Tidal Observations. 283 



the expression becomes 



h = A n cos F* + S m sin F m + R n cos [ V m - (o-w)t- J 

 + .R/cos [F + (F ar)*-], 



,] + B/COS [(<r ar)*- 

 + sin F{B m -HE w sin [ (a -nr) * + ]-, sin [(-)*-&]}. 



Hence, taking into account the equation which expresses that h is 

 a maximum or minimum, and neglecting the variation of s p com- 

 pared with that of F,, we have 



h cos F, = A m + -Rco8 [(--)*+ ,]+.22f cos [(<r nr) 

 & sin F = B m + R n sin [( w)*+] RI sin [()* 



The mean interval between each tide and the next is 6'210 hours. 

 Then if e be the increment of s p in that period (so that with a w 

 equal to 0' 5443 7 per hour, e is equal to 3 0> 3807), and if a, b be the 

 values of(<r w)t + and (a -nr) ^ at the time of the first tide under 

 consideration, the equations corresponding to the (r + l) lh tide are 

 approximately 



h cos V m -Am + RH cos (a + re) + HI cos (6 + re) , ~) 



L... (9). 

 A sin Fw = J5 OT + JBsin(a + re) J?;sin (6 + re) J 



If we take the mean of n -h 1 successive tides, the two latter terms 



on the right of (9) will be multiplied by " . and the r in the 



J O + l)sm3e' 



arguments a + re, 6-f-re, will bo equal to \n. It' the (w + 2) tb tide 

 falls exactly a semi-lunar-auomalistic period later than the first, 

 (w + l)e = TT. On account of the incommensurability of the angular 

 velocity a -or 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. 



Then, since e is a small angle, 



(n + 1) sin \e = *-, and sin ^(n+l)e = 1 ; 



hence the factor is 2/7r. 



Again ^ne = ^ir %e ; thus, if n + 1 is the mean number of tides 

 in a semi- anomalistic period, our mean equations are 



o/ ,- l , 

 ( n -f- j. ) 



^ (10), 

 -Rco6 (a^e)Ricos (i -e), 



x 2 



