496 "wanrOuT == 1868, 
angular velocities small in comparison with that of the earth’s rotation, so 
that the diurnal period of each relatively to the earth is something not very 
different from the lunar or solar twenty-four hours. Each one of the 
approximately semidiurnal tides ($ 2) is produced by one alone of these 
fictitious stars. 
13. One of the fictitious stars is what is commonly called in England the 
«mean sun,” being that point of the celestial sphere in the plane of the earth’s 
equator whose hour-angle is equal to mean solar time. For brevity we shall 
call it S. Another of them might be the ‘mean moon ” similarly defined 
(called M); but, to allow the same Tables ($$ 14, 15) to be used for the 
reduction of tidal observations of different years, we shall take it as a point 
moving in the plane of the earth’s equator, with an angular velocity equal 
to the mean angular velocity of the moon, and set for each year so that 
its hour-angle is 350°40 at 0”, January 1. Four others complete the 
number of the fictitious stars to be used in this Report*; they will be 
designated K, L, N, O. 
K might be the first point of Aries, but, for the same reason, will be taken 
as a point in the plane of the earth’s equator, with constant right ascension 
for each year, such that its hour-angle is 6°-25 at 0*, January 1. 
O is a fictitious star whose right ascension increases twice as fast as that 
of the mean moon, and which is so set for each year that its hour-angle is 
334°-54 at 0", January 1. L and N are fictitious stars whose rates of in- 
crease of right ascension are respectively greater than, and less than, that of 
the mean moon, by a difference equal to half that of the mean moon rela- 
tively to her perigee. Thus, if the motion of the moon’s perigee were 
neglected, the rate of increase of right ascension of L would be half that of 
N, and the arithmetic mean of that of N and O respectively; or, as the 
right ascension of K is constant, the rates of increase of right ascension of 
L, M, N, O would be equi-different. 
14. Thus, according to the preceding notation, the rates of increase of 
right ascension of K, L, M, N, O are respectively 
0, d(6+z), «@, 1(30—@), 20. 
Relatively to the meridian of the earth, their angular velocities are— 
Y y—20—-29, y—9, y—Fot2a, y—2e. 
Unless the contrary is stated we shall always suppose these angular velocities 
to be reckoned in degrees and decimals of a degree per mean solar hour. 
Thus reckoned their values are— 
o='549015, y=15-04108, y—Jo—}a=14-76425, y—o=14-49206, 
—8o4+3@=14-21988, y—20=13-94305. 
Tf ¢ denote time reckoned in mean solar hours from the 0" January 1 of the 
ear, 
ce aC ES 
will be the hour-angles of the fictitious stars. These have been calculated by 
successive additions for each integral mean solar hour of the year, and 
subtraction of 360 every time a number exceeding 360 has been reached ; 
and the results have been tabulated. Preceding each hour-angle, the number 
which, multiplied by 15, most nearly agrees with it has been written. The 
following is a specimen page for two days of the year, of the Table thus 
formed :-— 
* Others may be introduced for the lunar elliptic diurnal, and for the declinational 
and elliptic disturbances of the solar tide, not yet fully investigated. 
—_—_ 
