36 



Ol- TIIK CENTRE OF IXERTIA, AND OF MOMENTUM. 



the single panicles of both equally, increas- 

 ing or diminishing tlieir distance. 



ii68. Definition. Tiie centre of inertia 

 of Ivvo boch'es is that point in the right line 

 joining them, which divides it reciprocally 

 in the ratio of their magnitudes. 



269. Theorem. The centre of inertia of 

 two bodies^ initially at rest in any space, re- 

 mains at rest, notwithstanding anj' reciprocal 

 action of the bodies. 



C K Suppose the bodie 



A / JK~ ~L — — ^ C- ]5 equalj and consisting 



each of a single parti- 

 T) ^ cle, then it is obvious 



that both will be equally moved by any reciprocal action, 

 and the centre of inertia will still bisect their distance (2 1 7) • 

 Again, let one body A be double the other B, and suppose A 

 to be divided into two points placed very near each other, as 

 C, D. Join BC, BD, take any equal distances CE, DF, BG, 

 BH, and they will represent the mutual actions of B on C 

 and D, and of C and D on B, and the motions produced by 

 these equal actions ; complete the parallelogram BGIH, 

 and the diagonal Bl will be the joint result of the motions 

 of B; which, when C and D coincide in A or K, becomes 

 equal to iBG, 2CE, or 2AK ; but L being the centj-e of in- 

 ertia, BLnaAL (268), therefore IL remains equal to 2KL 

 (15), and L is still the centre of inertia. And in the same 

 manner the theorem may be proved when the bodies are in 

 any other proportion. 



270. Definition. The joint ratio of the 

 masses and velocities of any two bodie^is the 

 ratio of their momenta. 



271. Theorem. Tlie momentum of any 

 body is the true measure of the quantity of 

 its motion. 



For the same reciprocal action produces in a double body 

 half the velocity, the common centre of inertia remaining 

 at rest (260) ; and, the cause being the same, the effects 

 itiust be considered as equal : and when the reciprocal 

 force varies, the velocity of both bodies varies in the same 

 ratio. 



272. Definition. The centre of inertia 

 of three or more bodies is found by consi- 

 dering the first and st cond as a single body, 

 equal to their sum, and placed in their com- 



mon centre of inertia, determining the cen- 

 tre of inertia of this imaginary body and the 

 thud, and proceeding in the same manner 

 for any greater number of bodies. 



273. Theorem, The centre of inertia of 

 three or more bodies will be the same by 

 whatever steps it be determined. 



Let a, b, and c, denote the 

 masses of the three bodies A, B, 

 and C ; let D be the centre of 

 inertia of A and B, and take ED : 

 EC : ; c : a + b ; draw AEF, then 

 F will be the centre of inertia of 

 B and C, and AE to EF as b+c to a. Draw DG and FH 

 parallel toBC and BA, then (121) AD ; AS : : DG : BFr: 



DG.— =DG.^i^ (32); and DE : EC : : DG : CFziDG. 



^=DG.f±i, therefore BF: OF: :i:i 

 DE c be 



; c : b, and F is 



b b 



FH=BD. , but AD=BD.-, and FH : DA ; 



b-{-c a 



the centre of inertia of B and C. Again, CB ; OF : : BD : 



b i 



: : a : b+c : : FE : EA (l"-2l), and E is the same point as if 

 determined from A and F. And from this demonstration 

 the proposition may be shown to be true in cases where the 

 number is greater, following the changes step by step. For 

 instance, that in 4 bodies the order 1,2,3, 4, will give the 

 same result as 3, 1, 4, 2 ; since (1, 2, 3), 4, is shown to be 

 the same as (3, 1, 2), 4 ; and (3, 1), 2, 4, the same as (a, 

 1), 4, 2, or 3, 1,4,2. 



274. Theorem, The velocit)' and direc- 

 tion of the motion of the centre of inertia of 

 any system of bodies, are the same as those 

 of a single body equal to their sum, to which 

 momenta equal to those of the several bodies, 

 and in parallel directions, are communicated 

 at the same time. 



Let A be the common centre of 

 inertia of B, C, and D, and E the 

 centre of inertia of C and D. Let 

 B move in a given time to F; then 

 joining EF, and drawing AG pa- 

 rallel to BF, G will now be the 

 common centre of inertia ; but 

 BF : AG : : AB+AE : AE : : B+C+D : B ; therefore the 

 momentum of the single body B+C+D iu describing AG 



