OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. 93 



Suppose that P is the point in the vertical 

 diameter through which the plane of division 

 passes, separating the sphere into the seg- 

 ments DAE and DBE, sustaining equal pres- 

 sures on their convex surfaces. 



Bisect A p and B P in the points m and n ; 

 then are m and n y the points thus determined, 



respectively the centres of gravity of the surfaces of the spheric seg- 

 ments DAE and DBE,* and Am, A.n are their perpendicular depths 

 below FG the horizontal surface of the fluid. 



Put D AB, the vertical diameter of the sphere or globe A DBE, 



d=:Am, the depth of the centre of gravity of the surface of the 



upper segment DAE, 

 zz A n, the depth of the centre of gravity of the surface of the 



lower segment DBE, 

 S zz the surface of the upper segment, 

 P zn the pressure upon it, 

 &'zz the surface of the lower segment, 

 p the pressure upon it, 

 s zz the specific gravity of the fluid, 

 x zz AP, the perpendicular depth of the point through which the 



plane of division passes, and 

 7TZZ3.1416, the circumference of the circle whose diameter is 



unity. 



Then, according to the principles of mensuration, the convex sur- 

 face of a spheric segment : 



Is equal to the circumference of the sphere, drawn into the 

 versed sine or height of the segment ivhose surface is sought. 



And moreover, the circumference of a sphere, or the circumference 

 of any of its great circles : 



Is equal to the diameter multiplied by the constant quantity 

 TT, or the number 3.1416. 



consequently, the convex surface of the upper segment DAE, is 



and that of the lower segment DBE, is 

 S zz D TT (D x) zz 3.1416 D (D x}. 



* It is demonstrated by the writers on mechanics, that the centre of gravity of 

 the surface of a spheric segment, is at the middle of its versed sine or height. 



