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14 U. S. COAST AND GEODETIC SURVEY. 



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. The mean longitude of a body moving 

 in an inclosed 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 T , which is common to both circles. In order 

 to have an equivalent origin in the moon's orbit, we may lay off 

 an arc ^ t' (see fig. 6) in the moon's orbit equal to ,0, T in the 

 ecliptic and for convenience call the point T ' 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 longi- 

 tude. 



Let us now examine more closely the spherical triangle ^ T ^ 

 in Figure 6. The angles co and i are very nearly constant for long 

 periods of time and have already been explained. The side Si '¥', 

 usually designated by N, 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 19 je&rs, 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 /, the 

 supplement of the angle Q, A T , has an important effect upon 

 both the range and time of the tide, which will be noted later. The 

 side A T, designated by v, is the right ascensicn cr longitude in 

 the celestial equator of the intersection A. The arc designated by 

 ^ is equal to the side ^ T — side Q, A and is the longitude in the 

 moon's orbit of the intersection A. Since the angles i and oo are 

 assumed to be ccnstant, the values of I, v, and ^ vdll depend directly 

 upcn iV, the longitude of the moon's ncde, and may be readily 

 obtained b}'' the ordinary solution of the spherical triangle ^ T ^. 

 Table 6 gives the values of /, v, and ^ for each degree of N. In the 

 computation of this table the value of co 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 

 the epoch taken as the basis of the computation would have resulted 

 in differences of less than 0.02 of a degree in the tabular values. 

 The table may therefore be used without material error for reductions 

 pertaining to any modern time. 



Looking again at Figure 6, it will be noted that when the longitude 

 of the moon's node is zero the value of the inclination / will equal the 



