146 RECORD AND RESULTS OF 



Half-monthly Inequality. — The theoretical formula for the half-monthly mequality 

 in time is, according to the equilibrium theory, 



h' + 7i cos 2p 

 where h and h' represent the elevations of the spheroid due to the sun and moon 

 respectively, <p the angular distance of the moon from the sun, and 6' the angular 

 distance of the pole of the spheroid (or of high water) from the moon's place. In 

 reality, however, the pole of this spheroid follows the moon at a certain distance, 

 the mean value X of which is known as the "mean establishment," and which cor- 

 responds to a distance of the sun and moon of ^ — a instead of ^. This retroposition 

 of the tide, which is mostly the effect of friction, has been called the "age" of 

 the tide. 



The above formula, in conformity with the wave theory, then assumes the form 



tan 2 (B'-X) =- ^^ "^^^ ^ (.?.-a) 

 ^ ^ h' + h cos 2 {p-a) 



the mean establishment ?u', the ratio of the solar and lunar effect — and the angle 



7i' 



of retardation a are to be determined from the observations. 



The theoretical expression for the half-monthly inequality in JieigJtt is, according 



to the equilibrium theory, 



2/ = J (7i'2 + 7r + 2h' h cos 2p) 



where y represents the height of the pole of the equilibrium spheroid above the 

 undisturbed mean level of the surface, this expression must be changed, in accord- 

 ance with the wave theory, into the following^ 



y= J(jr- + li" + 2 h' h cos 2 (cp—a)\ 



the values of 7i', h and a must be found from the observations. 



In order to compare our observations with these theoretical expressions the luni^ 

 tidal intervals and heights of Table II were first arranged according to the time of 

 the moon's transit ; the total number of observations being comparatively small, the 

 results by the two series were at once united, for which purpose the heights of the 

 second series were all diminished by 1.2 foot to reduce them to the same plane of 

 reference. No distmction was made between upper and lower transits. For the 

 high waters as weU as for low waters twelve groups of lunitidal intervals and cor- 

 respondmg heights were formed, and the values of each group, extending over one 

 hour, were united into a mean, of which process the following is an example : — 



> Art. (535) Tides and Waves. tan 2 (e — x) = ^" ^'" ^ '^"^ ~ ITI"-^ and 



31" + S" cos 2 (m — s - a) 



y = Um"'' + 2 M'" S'" cos 2 (jTi^IZs —a) + >S'"=) 



