Maj. Barnard on the Dynamic Theory of the Tides. 355 
Let aa! o/b be an elementary area of ocean 
lace, the angular co-ordinates of a being # 
and 6, and ab and aa’, being equal to da and 
§6, Let the depth of the water at a be 7. Fe 
According to the assumption with regard to 
the i alluded to before, 7 varies with the 
oa [J 
R 8 is 
past the point a with a velocity which would ag he in that 
ime through the space u, and past the point a’, t 
Space u-+- 38, 
im area of the section ab is 7m, and of the section a’d’, is 
+7526) (1+ cot@36)3a, (Since it is easily seen that the side 
aH is to ab (or dm) as sin(6+-06):sin8: or as 1+ cot0d8:1 
(nearly ). Therefore the quantity of water which flows through 
the section @& in the short time ¢, will be uy, and through t 
Section a’b’ will be (u+ pe 0) G42 36) (1-+4+-cot004) da, The 
diference between these uantities, is the quantity of water sub- 
tracted from (or added tc) the ocean area aa! b/d, and is (omitting 
(Mantities of the 3d order) 
mae 7800 at wy cot 0d08a. 
The area of the section aa’ 8B is (nearly) 308% (for conveniete, 
ese an ication by the radius of the earth is omitted with all 
Pps angular quantities, as it does not affect the results) a . 
lial (or rise) of the tide w, due to the southern component 
iui! motion, will evidently be equal to the foregoing expression 
mided by the area 363-5; aw 
If we now consider the eastern component of velocity dv’ the 
Wantity of water which ru eastwardly through the section aa’ 
Whose area is 30 in the Fes tis 708, and the quantity which 
*ovs through 6d’ (since both 7 and 94 are constant int diree- 
: t d : 
‘ ‘a and dividing by 908 we get the actual total (or 
7 Sh€ tide, : 
oe. 
: 
. 
ee 5 
