154: W. D. Lambert — Mechanical Curiosities 



part of the great-circle section of the spherical surfaces 

 that lie below the ^ ^water-line ' ' ; the element of area dy dz 

 is to be treated as positive. The fact that the portions of 

 the '^water-line'' on the two sides of the ^2;-plane do not 

 have quite the same projection upon it will be dealt with 

 later. Neglecting the difference between the two projec- 

 tions we have from (1) 



Y,-Y^ = J,hxz. (7) 



The a; is, of course, the positive a; of P ; the z is essentially 

 negative. Thus it is seen that p^ is essentially positive, 

 since h is positive. It is evident that this must be so, for 

 gravity is greater at any given depth on the poleward 

 side of the sphere than on the equatorward side, so 

 that pressure must be greater also. 

 Using (7) we get 



p, 



h p j j zx dy dz. (8) 



It may be noted that x dy ds is the volume of an elemen- 

 tary prism and that zx dy dz is the moment of this prism 

 with respect to the a^^-plane. If we call z^ the depth 

 below the a;2/-plane of the center of gravity of the volume 

 considered, and fs the entire volume itself (on both sides 

 of the 2/^-plane) 



p^ = — 2 h p Vs z^ (9) 



The volume v^ is nearly the entire volume of the sub- 

 merged portion of the sphere, differing from it by the 

 small volume between the surface formed by the projec- 

 tion lines of the ^'water-line" and the surface of the geoid. 

 This small volume is zero in the ordinary theory of float- 

 ing bodies which treats the free surface as plane. The 

 quantity Zs corresponds to the depth of the center of 

 buoyancy in the ordinary theory. 



The pull of gravity on the sphere is not exactly along 

 the 0-axis, since the direction of gravity at the center C 

 is not quite the same as at the origin. The ^-component 

 of the pn^l is the mass of the sphere multiplied by the 



value of -^ at the center, or calling ^xthe pull of gravity, 



V the volume of the sphere, and C the ^-coordinate of 

 its center, we get 



^, = 2crvhl (10) 



