ON HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 325 



elliptic, evectional and variational inequalities in her distance and motion. Tliese 

 inequalities have speeds 



<r — w, 2(<r — xo), o- — 27) + ■or, 2((r - tj), 



where ir denotes the mean motion of the lunar perigee and r) that of the sun. The 

 effect of each is to make the moon's sidereal motion increase and decrease with the 

 reciprocal of her distance, and thus to make the period of the tides increase and 

 decrease with their range. 



The effect of the first order elliptic inequality and the evectional inequality is 

 the introduction of new harmonic constituents of speeds 



2(7 - <r) ± (o- - m), 2(7 - <t) ± (<t - 2ri + w) 



of which, for the reason just given, the greater are those of speeds 



2(7 - (t) - (0- - m), 2(7 - <r) - (<r - 2?) + rsr). 



The effect of the second order elliptic and variational inequalities is sufficiently 

 represented by the introduction of new harmonic constituents of speeds 



2(7 - <r) - 2((r - -Br), 2(7 - ff) - 2(ff - r,). 



The daily mean level of the water depends slightly on the departure from spheri- 

 city of the spheroid, so that we have long-period elliptic, evectional, and variational 

 constituents of speeds, 



(T — w, a- — 2r) + tsr, 2((t — tj), 



respectively. 



.3. If the moon moved with constant angular speed in a parallel of latitude other 

 than the equator, consecutive high tides would be unequal except at the equator, 

 and we should require the introduction of a new constituent of speed 



7-0", 



with an amplitude vanishing at the equator. Also, the amplitude of the constituent 

 of speed 2(7-0-) would be less than when the moon was in the equator. 



But since the declination of the moon changes, the diurnal constituent requires 

 modiHcation. If its amplitude could be regarded as changing harmonically with 

 speed <r, it would be replaced by two harmonic constituents of equal amplitudes and 

 speeds 



y — (T ± ff. 



Owing to the fact that this is not quite so, the amplitude of the constituent of 

 speed 7 — 2ir is a little greater than that of speed 7, and there is another smaller 

 constituent of speed 



7 -I- 2<r. 



Again, introducing the first order elliptic inequality we get new harmonic con- 

 stituents of speeds 



(7 — 2<r) ± (ff — w), 7 ± (er — or), 



of which those of speeds 



7 — tr ± or 



are regarded as forming a single constituent of speed 



7 — (T 



and slowly varying amplitude. The second order elliptic, the evectional and varia- 

 tional inequalities give rise to new constituents of speeds 



(7 - 2(r) - 2((r - nr), (7 - 2<r) - (<r - 2?) -1- tr), (7 - 2(7) - 2(«r - ij). 



Also, the changing declination of the moon causes the amplitudes of the semi- 

 diurnal constituents to vary, but it is sufficiently accurate to take mean values in 

 all cases except that of speed 2(7 — <r). As the effect is to make the speed and 

 range of tide increase or decrease together, we get a new constituent of speed 



2(7 - (r) + 2<r. 



