136 MOTION OF FLUIDS. [Book L
is of the same order. If then equation (77), be muhiplied by driu
integral will be
285. Since this equation has been integrated with regard to r
only, X must be a function of 0, tsr, and <, independent of r, accofding
to the theory of partial equations. And as the function in r is of the
order — it may be omitted; and tlicn
r
f
by wltich equation (70) becomes
'■'•{(■^)-'«'"""-(^}-
+ r«ScT {sin* (-^^ + 2/i sin cos ^(—^ = X^-
2S6. But as S does not contain r, t, or y^ it is independent of the
depth of the ]>articlo; liencc this equation is the same for a particle
at the surface, or in its neighhourliood, consequently it must coincide
with equation (76); and therefore
SX = JF' - giy.
2h7. Thus it appears, that the whole theory of the tides would be
determined if integrals of tlic equations
rots
{sin« ((!!L^l + 2;/ sin 6 cos f^ = — gXy + 8P
y-. _ djyi) _ d( 7r) _ 7!i
cosO
d6 dxsj sin.
could be found, for the horizontal flow might be obtained from the
first, by making the coeflicients of the independent quantities iO,
ScT, separately zero, then the height to which they rise would be
found from tlic second. This has not yet been done, as none of the
known methods of analysis have hitherto succeeded.
28S. These ecpiations have been formed on the hypothesis of the
'larth being entirely covered by the sea; hence the integrals, if they
