64 Statics. 



*ig. 48. JJQ^ p or a seconc [ example, we take the sphere. We now 



have y = y/q x z 2 ,(a being the diameter] andvra<tiu*x3 = 



-v . ^ adx 



Cal, 98. 



therefore^ 2 7t(h x)y Vdx 2 + dy* will become 







and this expression, C being the centre, which gives h \ a, will 

 be equal to J' n a (| <z d a? a^ c^ a?), which being integrated, is 

 n a (| a x x 2 ) or n a x ( a \ x). Now we have found 

 Cal 98 ^ at tne SUI *f ace tf of the spherical segment AMLMA, was n a x ; 

 we have, therefore, for the distance CG of C, the centra of the 

 sphere, from the centre of gravity G = 



that is, the centre of gravity G is in the middle of the altitude APof 

 this segment. Hence we derive the general conclusion, that the 

 centre of gravity of the surface of a spherical zone, comprehended 

 between two parallel planes, is the middle point of the altitude of this 

 zone. 



111. We shall terminate this branch of our subject, with 

 the investigation of the centres of gravity of solids. 

 Fig. 46. jf we cons id er a solid as made up of laminae infinitely thin, 

 and parallel to each other, and represent generally by tf, one 

 of the opposite bases of each lamina, and by d x its thickness, 

 we shall have tf d x as the expression for each lamina ; and 

 consequently <5 (h x) d x for its moment with respect to a 

 plane A [parallel to these laminae^ whose distance AC from the 

 vertex A we represent by h. Therefore, denoting by b the 

 bulk ALMMA, the distance of the centre of gravity from C will 



be = ~ ^ ~ x ' . Now the value of 6 is determined 







Cal 100 ^7 methods which have been heretofore given, and that of 

 &c. J 'd (h x) dx is found by the same methods, when the 

 value of 6 is known in terms of x. We shall thus obtain 

 the distance of the centre of gravity from a known plane. 

 We might find in the same way the distance of this centre 

 from eack of the two other planes, perpendicular to one another, 



