ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 5 



12. Again, let a, b and c, be a system of three very small bodies or 

 particles of matter, any how situated 

 with respect to the plane A B c D, and 

 connected together by the straight 

 lines a b, be and a c ; and suppose 

 the two bodies a and b to be collected 

 into their common centre of gravity 

 at the point m. 



Join the points m and c by the 

 straight line me, and let n be the place of the common centre of 

 gravity of the three bodies a, b and c ; draw the lines mq,nu and 

 cv parallel to each other, and respectively perpendicular to the plane 

 A B c D; join qv, and because the points m, n and c are situated in 

 the straight line m c ; it follows, that the points q, u and v must also 

 occur in a straight line ; consequently, q v will pass through the 

 point u. 



Through n, the common centre of gravity of the three bodies o, b 

 and c, draw st parallel to qv, meeting mq in t and vc produced in 

 s ; then are the triangles m n t and ens similar to one another ; but 

 by the property of the lever, and because the body at m is equal to 

 a 4- 6, we obtain 



a ~\- b : c : : c n : m w, 

 and by similar triangles, we have 



c n : m n : : c s : m t ; 

 therefore, by the equality of ratios, we get 



a -|- b : c : : c s : m t ; 



consequently, by equating the products of the extreme and mean 

 terms, we shall obtain 



(a -f b) X m t zz c X c s ; 

 now mt mq t q, and c sv s vc; hence we get 



(a -f- b) (mq tq) c(ysvc). 



But it is manifest by the construction, that t q and v s are each of 

 them equal to nu ; therefore, by substitution we have 



(a -|- b (mq n ) zr c (nu v c) ; 

 therefore, by collecting the terms and transposing, we get 



(a + b + c) Xnu=(a+b) Xmq + cXvc; 

 now, it has already been shown in the case of two bodies, that 

 a X^a-f b X rb 



n <*= ^+T~ 



therefore, by substituting this value of m q in the step immediately 

 preceding, we shall obtain 



