408 OF THE CENTRE OF PRESSURE. 



dXed* Xsm.de s e?Xec?X/wXsi 

 therefore, the accumulated effect of all the forces, to turn the plane 

 round rw, must be proportional to the sum of 



{dXed*}Xs\n.des /mXsum of {dXed}Xsin-des, 

 and this by the definition is equal to nothing; hence we get 

 yVwXsum of {c?Xec?}sin.rfeszzsum of {dXed*} sin.e?es ; 

 therefore, by division, we obtain 

 _ sum o 

 ~ su 



But the sum of {dXed}, is obviously the same as the body, or 

 sum of all the constituent particles, multiplied into the distance of 

 the common centre of gravity ; and therefore, by the principles of 

 mechanics, fm is also the distance of the centre of percussion, if ef, 

 the common intersection of the plane with the fluid, be considered as 

 the axis of suspension, the plane being supposed to vibrate flat-ways. 

 By reasoning in the same manner as above, it will readily appear, 

 that the effect or momentum of the pressure on c?, to turn the plane 

 about the ordinate PW, varies as 



dXe dXdnX$in.de s; 



but by subtraction, it is dn cd en ; therefore by substitution, the 

 force on d to turn the plane round PW, varies as 



and consequently, the effect of all the forces to turn the plane around 

 pn, must be proportional to the 



sum of {dxdeycd} siu.des crcXsum of {dxed} s'm.des ; 

 but by the definition, the sum of these forces is equal to nothing; for 

 the plane has no tendency to incline to either side, being sustained in 

 a state of quiescence by means of the fluid, and the equivalent oppos- 

 ing force applied at the centre of pressure ; hence we get 

 cwXsum of [dXed}~ sum of {dXdeXcd} ; 

 therefore, by division, we shall have 

 __sum of {dXdeXcd} 



sumof{dXde} ' (301). 



which expression also indicates the distance of the centre of percus- 

 sion ; from which it is manifest, that the centres of pressure and 

 percussion coincide, when the line of common section between the 

 plane or the plane produced, and the surface of the fluid is made the 

 axis of suspension. This being the case, it is evident, that the formulse 



