HARMONIC ANALYSIS AND PREDICTION OF TIDES a 
node.” Celestial longitude is always understood to be measured 
toward the east entirely around the circle. Longitude in the celestial 
equator reckoned from the vernal equinox is called right ascension, 
and the angular distance north or south of the celestial equator is 
called declination. 
22. The true longitude of any point referred to any great circle in 
the celestial sphere may be defined as the arc of that circle intercepted 
between the accepted origin and the projection of the point on the 
circle, the measurement being always eastward from the origin to the 
projection of the point. The true longitude of any point will generally 
be different when referred to different circles, although reckoned from 
a common origin; and the longitude of a body moving at a uniform rate _ 
of speed in one great circle will not have a uniform rate of change when 
referred to another great circle. 
23. The mean longitude of a body moving in a closed orbit and 
referred to any great circle may be defined as the longitude that would 
be attained by a point moving uniformly in the circle of reference at 
the same average angular velocity as that of the body and with the 
initial position of the point so taken that its mean longitude would be 
the same as the true longitude of the body at a certain selected position 
of that body in its orbit. With a common initial point, the mean 
longitude of a moving body will be the same in whatever circle it may 
be reckoned. Longitude in the ecliptic and in the celestial equator 
are usually reckoned from the vernal equinox Y, which is common to 
both circles. In order to have an equivalent origin in the moon’s 
orbit, we may lay off an arc & 1’ (fig. 1) in the moon’s orbit equal 
to £& Y in the ecliptic and for convenience call the point Y’ the 
referred equinox. ‘The mean longitude of any body, if reckoned from 
either the equinox or the referred equinox, will be the same in any of 
the three orbits represented. This will, of course, not be the case for 
the true longitude. 
24. Let us now examine more closely the spherical triangle 2 T A 
in figure 1. The angles w and 7 are very nearly constant for long 
periods of time and have already been explained. The side 27, 
usually designated by JN, is the longitude of the moon’s node and is 
undergoing a constant and practically uniform change due to the 
regression of the moon’s nodes. This westward movement of the 
node, by which it is carried completely around the ecliptic in a 
period of approximately 18.6 years, causes a constant change in the 
form of the triangle, the elements of which are of considerable im- 
portance in the present discussion. The value of the angle J, the 
supplement of the angle A 7, has an important effect upon both 
the range and time of the tide, which will be noted later. The side 
A 7, designated by »v, is the right ascension or longitude in the 
celestial equator of the intersection A. The arc designated by 
& is equal to the side 2 Y—side & A and is the longitude in the 
moon’s orbit of the intersection A. Since the angles 2 and w are 
assumed to be constant, the values of J, v, and & will depend directly 
upon NV, the longitude of the moon’s node, and may be readily 
obtained by the ordinary solution of the spherical triangle QT A. 
Table 6 give the values of J, v, and é for each degree of N. In the 
computation of this table the value of w for the beginning of the 
twentieth century was used. However, the secular change in the 
obliquity of the ecliptic is so slow that a difference of a century in 
