4 ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 



or particles of matter, supposed to be col- 

 lected into their respective centres of gravity, 

 and let A B c D be a smooth rectangular 

 plane or surface, placed in any position 

 with respect to the bodies a and b. 



Connect a and b by the straight line 

 ab, and let m be the place of their 



common centre of gravity; draw the straight lines a p, m q and 

 br respectively perpendicular to the plane A B c D, and consequently 

 parallel to one another ; join pr, then because the points a, m, b are 

 situated in a straight line, the points p, q, r are also in a straight line, 

 and therefore p r will pass through the point q. 



Through m t the common centre of gravity of the two bodies a and 

 b, draw st parallel to pr, meeting br in s, and pa produced in t ; 

 then the triangles ami and bms, are similar to one another; but 

 by the property of the lever, we have 



a : b :: bm : am, 

 and by similar triangles, it is 



bm : am : : bs : at; 

 therefore, by the equality of ratios, we obtain 



a : b : : bs : at ; 



from which, by equating the products of the extreme and mean terms, 

 we get 



a X at bx bs. 



Now, it is manifest by the construction, that at ptpa, and 

 b s r b r s; therefore, by substitution, we obtain 



a (p tpa)=b (rb rs)', 



but by reason of the parallels p r and t s, the lines p t and r s are 

 respectively equal to m q ; hence we have 



a (m q p a) m: b (r b m q), 



and from this, by collecting the terms and transposing, we get 

 (a + b) mq a X p a ' -f- b X r b, 

 and finally, by division, we obtain 

 a xa b X rb 



- - 



a -f- b 



COROL. Here then, the truth of the proposition is manifest with 

 respect to a system composed of only two bodies ; that is, 



The distance of the common centre of gravity from the 

 plane to which the bodies are referred, is equal to the sum of 

 the products, arising by multiplying each body into its dis- 

 tance from the given plane, divided by the sum of the bodies. 



