OF THE MOTION OF FLUIDS. 191 



= ^« + -27^(3008-'^- 1) - -g^-cos'e -f{t). 



When r is indefinitely great this equation becomes p=g^—f{t)', and 



as for this value of r the velocity = 0, p must = g% ; therefore /{t) = 0. 



If now we put r = a, and i8 = ss,, the co-ordinate of P, we obtain the 



?^* cos 20 

 pressure (/>,) at P, = gz, -\ . The portion of this resolved in 



the vertical direction is jo, x cos i FOB. But from the spherical 

 triangle PQB, cos /. FOB = cos w sin 9. Therefore the vertical pressure 

 is p, cos w sin 6. The element of the surface at F = ad9 x a sin ddw. 

 Hence the whole vertical pressure = //jOia'sin''^ cos wt/^c^w 



=ga^ff%i sin^OeoswdOdw + ——— ff sin^ 9 cos29 cos wd$dw. 



M 



The first term is plainly the weight of water displaced, and is there- 

 fore equal to — -(2«' — 3«'7 + 7*), the specific gravity of the water 

 being 1. The integrations with respect to u> must be taken from 

 a,= — cos"^ — jT—r to -f cos"^ — ^-;r , and the integrations with respect 



to 6 from sin"'— to the supplement of that arc. Between these limits 



of w, fcoswda) = 2\/i T ; and between the limits of 9, 



a'^sm''9 



2fsm'9cos29d9 x/i _ , '^^ =_ J fi _:>:!') . 

 •^ ^ «*sin''0 2 V «V 



Therefore if JV = the weight of the sphere, which is the same as the 

 whole vertical pressure, and w = the weight of fluid displaced, 



IV =w- 



4 



B B 2 



i^-i)- 



