VOL. XXVIII.] PHILOSOPHICAL TRANSACTIONS. Q 



line CO, and put the body abc = a. Thqn, from the known property of the 

 centre of gravity, it will be m = c^ X a. Hence co = — — . 



Prop. 1. Theor. 1 . The same things being supposed, let the point o be 

 found in the right line cg passing through the centre of gravity g (fig. 6).- 

 Then will the point o be the centre of oscillation of the body a. 



f O C^ (T tic 



For in this case it is — = -2- = c?; hence co = ( by Cor. of prop. 1 =) 



CO C tr Q ^ ■*■ 



— - — . But A is given, and the point c being given, cg and the quantity c 

 are given. Hence co is given, whatever be the inclination of the vibrating 

 body to the horizon. Therefore, by the definition and problem 1, o is the 

 centre of oscillation of the body a. a.E.D. 



Prop, 3. Theor. 1. The same things being supposed, let d be the aggre- 

 gate of all the Gz* X p. Then it will be co = cg H ^—, 



O ^ ' CGX A 



For to CG draw the perpendicular zp, fig. 7, and it will be cz^ = cg^ + gz^ 

 — 2CG X GP, when p falls between c and g. But when f falls in cg produced, 

 it will be cz'^ = cg^ + g-?'^ + ^cg X of. Therefore c = (aggregate of all 

 the cz^ X p =) aggregate of all the cg^ X p + gz^ X p — 2cg X gp X /) + 

 2cg X of X p- But, G being the centre of gravity, the aggregate of all the 

 2CG X GP X /) = the aggregate of all the 2cg X of X p' Therefore c = the 

 aggregate of all the cg^ X /) + gz^ X /) = cg"^ X (a + d). But, by theor. 1, 

 it is CO = . Therefore co = cg H •. a.E.D. 



CGX A • CGXA 



Coral. Hence the parallelogram cg X go is given. For 

 CO = . But A and d are given. Therefore cg X go = - is given. 



CG X A A- 



Prop. 4. Theor. 3. The same things being supposed, and having consti- 

 tuted the physical particle , which being actuated by its own gravity shall 



vibrate about the point c; the motion of the space abc shall be just the same, 

 as if it were actuated by the oscillation of the body a. 



This appears both from the nature of the centre of gravity, and by prob. 1 . 

 For S2ii-i is the aggreerate of all the f = -^. 



CO °° ° CO^ CO^ 



Prop. 5. Proh. 2. Having given the magnitude of any body a, with the 

 centre of gravity g, and the point of suspension c : to find its centre of 

 oscillation o. 



This is done either by theor. I , by finding the quantity c ; or by theor. 2, 

 by seeking the quantity d. 



Scholium. For performing the calculation in a particular case, the quantity 

 c or D is to be chosen, according as may be suggested by the nature of the 

 proposed figure. Then either of these being given, the other will be also given 



VOL. VI. C 



