Page:Somerville Mechanism of the heavens.djvu/224

'''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