60 



POSITION OF THE METACENTRE. 



28. The Position of the Metacentre ; the Measure 

 of the Stability. Since the stability of a body depends 

 principally upon the distance of the metacentre from the 

 centre of gravity of the body, it becomes important to de- 

 termine the position of the metacentre. 



Let A = the cross-section ABD = HKD (Fig. 23) of 

 the immersed part of the body (Art. 27), and A l = the 

 cross-section ACH = BCK ; let g and g' be the centres of 

 gravity of ACH and BCK ; let a = the horizontal distance 

 EL, between these centres of gravity, and s = the horizon- 

 tal distance between and 0', the centres of buoyancy. 

 Then, taking moments round Q, we have, 



HKD x MN - ACH x RN = ABD x NT + BCK x NL ; 



or, 



A (MN - NT) = AI (RN + NL); 

 .*. As = Aa', 



or, 



and 



OM = 



00' _^i_ 

 sin 6 A sin 0' 



which is the height of the metacentre above the centre 

 of buoyancy. 



Let GO = e\ then 



c = GM = e + -Jr^-v 

 A sin 



(1) 



which gives the height of the metacentre above the cen- 

 tre of gravity. 



Substituting this value of c in (1) of Art. 27, we get 



which is the measure of the stability. 



