308 CELESTIAL iMECHANlCS. 1. vil. 36. 



have, for the fluent of the equation Ozz—i — - + 9^ (_-4--_- 



-\- — : ) , taken with respect to r, = r^s — (i^"s) + r^y 



sin 9 / 



/dw ,dv u cos 9\ . . . 1 i- A 



(-—+—- + — : , since y is the particular value of /dr 



\d9 dw sin 6 / ^ -^ 



between these limits. The quantity r^s — (rh) is also very 



nearly equal to r^ ) s — (5) J- + 2ry (s), (5) being the value 



of 5 at the bottom of the sea, and considering the minute- 

 ness of 7 and of 5, the latter part of this expression may be 

 neglected in comparison with the former, and we may call 



r^s — {t^^s) — r-^ s — (s) >. Now the depth of the sea, 



corresponding to the angles 9-{-au, and nt + 'sr + av, is y + a 



} s — (s) i : and if we consider the angles 6 and " nt + " 



'sr as beginning at a fixed point and a fixed meridian on 

 the surface of the earth, which will soon appear to be ad- 



Cl'Y d v 



missible, this depth will be 7 + aw -—-\-av-—, besides the 



d5 d-sr 



elevation ay of the particle above the surface of equili- 

 brium, [for since 7 is, by the supposition, a function of fl 

 and Tsr, it is necessary to comprehend in the equation its 

 variations dependent on those of these angles;] conse- 



Qy (J<y 



quently s — (.9)zi?y-f w -7""^ ^* T~* '^^^ equation^of the con- 



tinuity of the fluid will therefore become y =■ — — 



d(yv) yu cos fl dw dy du dy 



d-ar sin 9 dd d5 d'si d-sr 



yu COS 9 . ^ ^ /dw , du , M cos 9\ dy 



% — ; since 5— (5)=: — y(-;- + — + — -. )z::v+w — 



sm d ^ ' ^Vdfl d'3r sind / ^ dd 



dy 

 -f V -~ , which amounts to the samel 

 d^ 



