296 REPORT— 1876. 



place. Taking the actual figure of the earth's sea-surface, we must subtract 

 a certain positive or negative quantity a from the radius of the spheroid that 

 would bound the water were there no land, a being determined, according to 

 the moon's position, to fulfil the condition that the volume of the water 

 remains unchanged, and being the same for aU points of the sea at the same 

 time. Many writers on the tides have overlooked this obvious and essential 

 principle ; indeed we know of only one sentence* hitherto published in which 

 any consciousness of it has been indicated. 



" The quantity « is a spherical harmonic function of the second order of the 

 moon's declination and hour-angle from the meridian of Greenwich, of which 

 the five constant coefficients depend merely on the configuration of land and 

 water, and may be easily estimated approximately by not very laborious qua- 

 dratures, with data derived from the inspection of good maps. 



" 808. Let as above 



r==a(l+u) (14) 



be the spheroidal level that would bound the water were the whole solid 

 covered ; u being given by (13) of § 804. Thus, if ffda denote surface in- 

 tegration over the whole surface of the sea, 



affudcr 



expresses the addition (positive or negative as the case may be) to the volume 

 required to let the water stand to this level everywhere. To do away 

 with this change of volume we must suppose the whole surface lowered 

 equally all over by such an amount « (positive or negative) as shall equalize 

 it. Hence if i2 be the whole area of sea, we have 



a-=.-^ (15). 



ii 



And v=r-a = a{\^u--^J^) . . . . \ (16) 



is the corrected equation of the level spheroidal surface of the sea. Hence 



h=a{u-'/M!L} (17), 



where h denotes the height of the uurface of the sea at any place above the 

 level which it would take if the moon were removed. 

 "To work out (15), put first, for brevity, 



Mo* 



and (13) becomes 



u = T(cos-d-i) (19). 



Now let I and \ be the geographical latitude and west longitude of the place 

 to which u corresponds ; and i|/ and S the moon's hour-angle from the meri- 

 dian of Greenwich, and her declination. As 6 is the moon's zenith distance 

 at the place (corrected for parallax), we have by spherical trigonometry 



cos 0=cos Z cos 2 cos (\ — )//) -t- sin ? sin 2 ; 

 which gives 



3cos20 — 1 = 



f cos=Z cos'g cos2(X -;//) + 6sin ?cos I sin t cos S cos(\ -^p) + i (3sin2S- l)(3sin-Z - 1) (20). 



"Rigidity of the Earth," § 17, Phil. Trans. 18G2. 



'=IW3 (18): 



