Astronomical and Nautical Collections. 311 
at once the height of the tides in any port, if the coefficients were 
sufficiently determined, and even without this determination affording 
some interesting conclusions from facts that are already well known. 
For a canal or a sea lying in an easterly and westerly direction, the periodi- 
cal force is shown to vary as sin cos Alé, sin Az., and for a canal deviating 
from that direction in a given angle, as sin cos Alt. sin (Az. + Dev.). And in 
the two cases of a canal running east and west in any latitude, and of a canal 
situated obliquely at or near the equator, the force becomes, still more simply, 
first, L sin cos Decl. sin Hor, Z + 1’ cos* Decl. sincos Hor. /, i being the 
sine of the Jatitude, and L’ its cosine; and secondly, if p be the sine of the 
deviation, or of the angle formed by the length of the canal with the equator, 
and D’ its cosine, D sin cos Decl. cos Hor. 7 + v’ cos? Decl. sin cos Hor. / 
A series is then found for representing the declination by means of arcs in- 
creasing uniformly with the time; but it is observed that for the purposes of 
calculation it is sufficient to suppose the sun and moon to move uniformly 
in the ecliptic, or even to have uniform motions in right ascension; whence 
-we obtain for the sun’s force, on a canal running east and west, putting @ for 
the sine of the obliquity of the ecliptic, © for the sun’s longitude, and ¢ forthe 
horary angle, S(t @ [ 5 cos (¢- ©) — }cos (¢ + ©)] +1e” [} cos(t—3 ©) 
—400s(¢+8@)J +1” [4cos(t—5 ©)—} cos ((+ 5 ©)] + 1 ( - 1%) 
‘ 2 a 
sin. 2 ¢ + a [isine(t+t@ +1sine(t-@)]: a’, @’, and e”, being 
coefficients derived from , and equal respectively to about .3645, .0078, and 
.00002, and a@? = 1585. From each of the terms, expressing the forces, the 
value of the corresponding portion of the space described may be obtained by 
means of the general Theorem K, substituting, in the case of the solar tide, for 
the coefficient of the simple resistance A, the value 4’ = 4 + 2.88 DM’, in 
which D is the coefficient of the resistance varying as the square of the velo- 
city, and M’ the supposed actual extent of the lunar tide; and for the lunar 
tide A” = A+ 2.88 DS’ + .3434 D(M’— S’). 
But without calculating the precise amount of all the coefficients, the author 
proceeds to demonstrate in general, that ‘‘ the results, with regard to the space 
described, will not differ much from the proportion of the forces, except when 
their periods approach nearly to that of the spontaneous oscillation, repre- 
sented by B.” And “ considering in this simple point of view the correct 
expression of the force ; we may observe that the phenomena, for each lumi- 
nary, will be arranged in two principal divisions, the most considerable being 
represented by } (1’, D’) cos* Decl. sin 2 Hor. Z, and giving a tide every 
twelve hours, which varies in magnitude as the square of the cosine of the de- 
clination varies, increasing and diminishing twice a year, being also propor- 
tional to the cosine of the latitude of the place or of the inclination of the 
