Or 
DISCUSSION OF TIDES IN BOSTON HARBOR. 
5. If we likewise put 
; 2’— the potential of the sun’s disturbing force, 
and let »’, rv’, »’, and v’ denote the same with regard to the sun, which the same letters without an 
accent do with regard to the moon, we shall have 
(4) 0/ = ¥,N’, cos s (nt+o0— y’) 
in which the expressions of N’, are the same as those of N, with v, 7, , and v accented. 
In the preceding expressions the origin of ¢ must be such as to make the angle nt-+a—y or 
nt+o—y’ equal to iz, that is, some multiple of z, when the moon or sun is on the meridian of the 
place with regard to which the force is considered. The values of s may be 0, 1, 2, &c., but there 
are no terms producing sensible effects in which s is greater than 3. 
6. In the comparison of tides with the forces producing them, it is necessary to either analyze 
the result and tide of the moon and sun in each port into its component parts, or to have the result- 
ant of the component forces of the moon and sun with which to compare them. The latter is pref- 
erable in tidal discussions and investigations, since the developed expressions of the resultant of 
the forces of the moon and sun being obtained, and all the constants accurately determined, these 
expressions, depending mostly upon celestial circumstances, serve, with a very few convenient mod- 
ifications, for every port; whereas if the former method is adopted, a troublesome analysis, and a 
determination of the constants belonging to each component of the tide, must be made for each port. 
By combining the preceding components, we get 
(5) Q+0'=3,VNZ+N2+2N, N% cos s () — 7) coss (nt +a—w-+ f) 
in which - 
N’, sin Ss (b — yp’) 
N, +N’, cos s (p—v’) 
(6) tan S ,= 
7. If we put 
Q,—the part of the preceding expression belonging to s, 
its development may be expressed in the following form: 
(7) 2.— CO, 3; P; cos 7; cos s (nt+G— yp + fz) 
in which the angles 7; and n¢-+-o—7-+/, do not increase exactly in proportion to the time on 
account of the variable motions of the moon and sun in their orbits, and the obliquity of the eclip- 
tic. The latter angle also varies with the changing value of @, which expresses the angle in right 
ascension between the moon and the position of a disturbing body which would represent the result- 
ant of that part of the forces of the moon and sun belonging to the characteristic s. In the pre- 
ceding expression ©, is the constant or average value of the coefficient of cos s(nt4+o—y-+ fs), 
and is independent of any of the inequalities. Its value depends upon terrestrial circumstances, 
and consequently is different in different ports. The constants P; and the angles 7; depend upon 
celestial circumstances only, and consequently are the same for every port. The constants P; are 
different for different values of s, and should be denoted by Piz, i) when it is necessary to distinguish 
them. LP, is the constant of integration and is equal to unity. ‘ 
8. We shall now give the angles and the numerical values of the constants, and also the mean 
values of the first derivatives of the angles in terms of the radius, belonging to the principal terms 
in the preceding expression of 2, for each value of s. In the expressions of the angles the following 
notation is used: 
» =the moows mean anomaly ; 
v= that of the sun; 
yg =the moon’s longitude; 
g'= that of the sun; 
w == the longitude of the moon’s ascending node. 
From the notation (§§ 4 and 5) we also have ; 
» — 1’ =the difference in right ascension between the moon and sun, usually expressed by the 
apparent time of moon’s transit. 
