OF THE MOTIONS OF FLUIDS. 303 



to l+Af (^+~ + -^j in article 365]. Hence we 

 obtain the equation 



0='-^^'+(f).l^+,-+-^j^)j+(f)-i;i>. [For 



f being = (f)+a/,f (f)+«f'|(l+aj+a^+a^\(r« + 

 2ars) sin (9+aa) = (f ) r- sin 9, (f) r^sin i+au cos «) — (j) t* 

 «in fl+aj.V sin 9 +(f) r"* sin 9 (a J+ a Jj+ a J^) + (f)2*r» 



sin9=0(140).0=^{,'^(,)(J4^+ii5^V(.)^'|-^ 



36. Cflwc o/* the motion of the sea, supposing it to he 

 deranged from the state of equilibrium hy the action of 

 very small forces. P. 101. 



375. Theorem. Retaining the notation 

 of the preceding propositions, and supposing 

 the sea of inconsiderable depth, we have, for 



the surface, i^^Q* ^"dF""" ^^ ^^^ ^^^ ^~It) "^ ^^^' 

 (sin *Q ~ + 2w sin cos Q ■^) = -gSy+SF'. (M) 



Since the density of the sea is uniform, we hare /=0, 



J ^, d(rr5) „ /dw , du , m cos 6\ ^ t^, 



and consequently -V-^+r2 (-r. +t- + — ^ir )=0. Now 

 dr VdS d-sr sm d / 



we may suppose the depth of the sea inconsiderable in com- 

 parison with the radius r of the terrestrial spheroid; and 

 calling this depth y, we may imagine y to be a very small 

 function of Q and ns, determined by the law of the depth. 

 If we consider the nature of the fluent of this equation 



