Astronomical and Nautical Collections. 313 
if B =.9, these angles become 45° and 70° 94’ respectively ; the difference in 
the latter case, 25° 24’, corresponding to a motion of more than 24 hours of the 
moon in her orbit. 
“ Tt appears then that, for ‘his simple reason only, if the supposed data 
were correct, the highest spring tides ought to be a DAY LATER than the con- 
junction and opposition of the luminaries ; so that this consideration obviously 
requires to be combined with that of the effect of a resistance proportional to 
the square of the velocity, which has already been shown to afford a more ge- 
neral explanation of the same phenomenon,” 
It may easily be admitted that this theory may require much further illus- 
tration, and perhaps discussion, before it can be rendered very popular, or in- 
telligible, in all its bearings; but in point of mathematical evidence, it may 
not be superfluous to insert here the reduction of the expression of the force 
acting on an oblique canal into the simple form which the author has adopted. 
without a demonstration, at the end of his paper. 
Since the force f = sin cos Alt. sin(Az. + Dev.) = sin cos Alé. (v’ sin Az. 
+ pcos 4z.); and sin Alt. = ut sin Decl. + 1’ cos Decl. cos Hor. Z 3 also 
cos Decl. sin Hor. 7 
cos Ale. 
Hor. 7 , and cos? Alt. sin? Az. = cos? Decl. sin? Hor. 7 =cos? Alt. (1— cos? Az.) 
and cos? Alt. cos? Az. = cos? Alt. — cos* Decl. sin? Hor. Z = 1— sin? Alé.— 
cos? Decl. sin? Hor. 7 ;, whence cos Alt. cos Az. = 1 — } (sin? Alt + cos? 
Decl. sin? Hor, Z) + 3 (sin? Alt. + cos? Decl. sin? Hor. Z)? — yg-++5 and 
finally, 
f = (“sin Decl. + v' cos Decl. cos Hor. /)(v' cos Decl. sin Hor. Z + D 
[1 — 4(sin? Alé. + cos? Decl. sin? Hor. Z) + 32....]) 3 which may readily 
be more completely developed if required. 
sin Az, = 3; we have cos Alt, sin Az. = cos Deel. sin 
But for a lake obliquely situated at the equator, when L = 0, and L’ = 1, 
the expression becomes sin Alt. = cos Decl. cos Hor. 7, and cos? Alt. cos? 
Az. = 1 — cos? Decl. cos? Hor. Z — cos? Decl. sin? Hor. Z =1 — cos? 
Decl. = sin? Decl., and cos Alt. cos Az. = sin Decl. ; whence 
f = cos Decl. cos Hor. Z (d’ cos Decl. sin Hor, Z + pv sin Deel.) = p’ cos? 
Decl. sin cos Hor. 7 + dD sin cos Decl. cos Hor. Z, which is the equation 
given at the end of the Article, agreeing with the equation of the form for a 
canal running east and west, in having for each luminary a semidiurnal tide 
which is greatest when the declination vanishes, and a diurnal tide increasing, 
onthe contrary, as the sine of twice the declination increases. 'The two for- 
mule give the same result for a canal coinciding with a part of the equator, 
and they appear in other cases to represent the force for every part of the same 
oblique great circle, the deviation at the equator being equal to the latitude 
when it becomes perpendicular to the meridian. 
Lapuace, assisted by the indefatigable Bouvarp, has lately published a very 
valuable continuation of his Researches on the Tides, as a XIIIth Book of his 
