1890.] Harmonic Analysis of Tidal Observations. 285 



Secondly, suppose the semi-lunar-anomalistic period indicated by 

 1 consists of ii-l-iii, that 2 consists of iv-f v, and so on. 



Obviously the result is got by writing t + \irl(<r~ w) for #, 

 or, what amounts to the same thing, by putting^' T in place of j ; 

 but we must also write S**" for S, so as to show that the summation 

 begins when (<r w)t is nearly equal to \TT. Then 



SHicos F = -B H cos (&-j) -Ri cos (ft +./), 1 



> ..... (14). 

 Si'A sin V M =-R* sin (,-./) -^ sin (ft +./). J 



Hence R* sin (ft, j) = SA cos 7 OT S* 11 ^ sin 7 m , 



BU cos (ft..;) = SA sin F S**& cos F OT , 



> ....... (15)- 



Ri sin (ft +;') = S/i cos F OT S*fc sin F 



cos (ft+j) = -S^ sin 7 S*fc cos F m . J 



These four equations give the four unknowns R*, , RI, ft, andy is 

 equal to 1'69 (<r w)t . 



Then if U H , ui denote the equilibrium arguments of the tides N ami 

 L at epoch, we have 



u n = fc -i 



where J> , 8 , p are the mean longitudes of moon, sun, and lunar 

 perigee at epoch, and v and are small angles, functions of the lon- 

 gitude of the moon's node (tabulated in Baird's Manual). 



Then if f, rt is the factor of reduction (also tabulated by Baird) for 

 the tides M 2 , N, L, 



KH = ft. + , Kl ft + Ui, 



In this investigation the interferences of the solar and diurnal 

 tides are neglected, on the assumption that they are completely 

 eliminated. 



The difference between a lunar period and an anomalistic period is 

 so small that the elimination of the diurnal tides will be satisfactory, 

 but the effect of the solar tide will probably be sensible, unless we 

 have under reduction 13 quarter-lunar-anomalistic periods, which 

 only exceed 6 semi-lunations by about 25 hours. 



The evaluation of the elliptic tides N and L from a series of obser- 

 vations shorter than a quarter year would be very unsatisfactory, and 



