C E N 



1>et A whole area, a area dut. temp., then 



~ a 



JSjt'. 1. In ellipses (force in the centre}, the periodic times ~ , and 

 are therefore equal in all ellipses. 



2 <r A C 

 Eo:. 2. In an ellipse (force in the focus). PT = _ *. 



CENTRE of Gravity. (t'ince, Play fair.) 



1. To find the centre of gravity of two given bodies, divide the distance 

 between them in the inverse ratio of their quantities of matter, and the 

 point so determined is the centre of gravity. 



2. To find the centre of gravity of any number of bodies placed in the 

 same straight line. 



Let the bodies be A, B, C, D, &c. and their distances from a given 

 point in the straight line be a, b, c, d y &c. then the distance of their centre 

 of gravity from this point is 



A a 4- B b 4- C r. 4. D d, & c. 

 A 4- B + C 4- D, *. 



3. In general, the distance of the centre of gravity of any system of 

 bodies from a given plane, is equal to the sum of the products of all the 

 masses, into their distances from the plane, divided by the sum of the 

 masses. 



Cor. If any of the bodies in this and the last Art. lie on the other side " 

 of the point or plane, their distances must be reckoned negative. 



4. Any number of bodies being given in position, to find their centre 

 of gravity. 



The bodies must be referred to three planes given in position, cutting 

 one another at right angles, one of them horizontal, and of course the 

 other two vertical. Let the bodies be A, B, C, D, their distances from 

 the given horizontal plane, a, b, c, d ; their distances from one of the ver- 

 tical planes a', b', <', d', and from the other a", b", c", d" ; then if we 



Aflr 4- B6 4- CP 4. D d 



A 4- B 4- {T+D 

 is in a horizontal plane at the distance a- from the given horizontal piano. 



Again take A" q "*" / -5ZL-l. and the centre of gravity Ls 

 A 4, ii 4- c 4- 1) 



in a plane parallel to the first of the two vertical planes, and distant from 

 ,it by the line A". Lastly, take in the intersection of these planes a point 



