TIDAL OBSERVATIONS. 299 



'< : 



811. The superposition of tlie solar semidiurnal on the lunar semidiurnal 

 tide has been investigated above (§ 60) as an example of the composition of 

 simple harmonic motions ; and the weU-knowu phenomena of the ' spring- 

 tides ' and ' neap-tides ' and of the ' priming ' and ' lagging ' have been 

 explained. We have now only to add that observation proves for almost all 

 places, whether oceanic islands or other open coast-stations, or in deep bays, 

 estuaries, or tidal rivers, the proportionate difference between the heights of 

 spring- tides and neap-tides, and the amount of the priming and lagging to be 

 much less than estimated in § 60 on the equilibrium hypothesis ; and to be 

 very different in different places, as we shall see in vol. ii. is to be expected 

 from the kinetic theory." 



The four lunar and solar diarnal and semidiurnal tides spoken of in 

 § 809 of the preceding extract are, in the harmonic analysis of this Com- 

 mittee, resolved into harmonic constituents with constant amplitudes and 

 epochs instead of the varying amplitudes and epochs which that statement 

 implies in virtue of the varying distances of the sun and moon from the 

 earth, and of the differences of their right ascensions from those of ideal 

 bodies moving uniformly in the plane of the earth's equator with constant 

 angular velocities equal to the mean angular velocities of the sun and moon 

 round the earth. 



To investigate, for either moon or sun alone, the equilibrium values of 

 these simple harmonic constituents, and to exhibit the simple harmonic ex- 

 pression for the long-period deelinational tide represented by the first line of 

 (238) § SO, call L the value of this line, D the value of the second line (or 

 the whole complex diurnal equilibrium tide), and S the value of the third 

 line (or semidiurnal equilibrium tide). 



Put 



sin I cos IcosX — C=P cos/, sin ? cos Z sin X — iB = F sin/ ; 



cos-Z cos2\ — ^ = G cos2 5', cos^Z sin 2\ — 33 =Gsin25r. 

 Then 



L=^arK(3sin=a-l), (I.) 



D=2ffrsinacosa.rcos(v/^-/), (II.) 



S=^cos2a.Gcos2(;^-5r), (III.) 



where F, /, G, g are constants for each place, having different values for 

 different places. Let ^ be the angle between the body's radius vector and 

 the ascending node of its orbit relatively to the earth's equator (which for 

 the case of the sun will be his longitude) ; let v be the right ascension of 

 this node (which for the case of the sun is of course zero) ; let a denote the 

 right ascension of the body reckoned from this node (which for the sun will 

 be his right ascension measured from the first point of Aries) ; let I denote 

 the inclination of the body's orbit to the plane of the earth's equator (which 

 for the case of the sun is nearly enough constant for our purposes and equal 

 to 23° 27' 19") ; lastly, let j( denote the sidereal time reduced to angle, that 

 is to say, the (Greenwich hour-angle of the first point of Aries. We have 



