52 



OF ROTATORT POWER. 



pul?e will be 1 — x; and Us momentum (i — t).()-(-i), 

 therefore the momentum is increased in the ratio of 

 1 to l+x, or in the subduplicate ratio of 1 to i+2r 

 +XX, which as x is diminished, approaches infinitely 

 near to the subduplicate ratio of 1 — x, to 1+x, or i 

 to l+2«+5!.r^ + 2.T^... since all the succeeding terms vanish 

 in comparison with the preceding : and in the same man- 

 ner it may be shown that at every succeeding step the mo- 

 mentum will be increased in the subduplicate ratio of the 

 bulk ; therefore the joint ratio of all the changes of momen- 

 . turn will be the subduplicate ratio of the corresponding 

 change of bulk. 



Scholium. The first body will also have a retrograde 

 motion after the collision, with the velocity x, and the subse- 

 quent bodies will recoil with velocities gradually smaller, in 

 the same proportion as their progressive velocities have been 

 smaller. If a second impulse be communicated to the first 

 body, it will impel the second with a velocity infinitely near 

 to that which the first impulse produces, and will itself re- 

 coil with a double velocity.* 



347- DF.riNiTioN. The product of the 

 mass of a body into the square of its velo- 

 city may properly be termed its energy. 



Scholium. This product has been called the living or 

 ascending force, since the height of vertical ascent is in pro- 

 portion to it ; and some have considered it as the true mea- 

 sure of the quantity of motion ; but .although this opinion 

 has been very universally rejected, yet the force thus esti- 

 mated well deserves a distinct denomination. 



348. Theorem. In two bodies perfectly 

 elastic, the joint energy, with respect to any 

 quiescent space, is unaltered by collision. 



Let the bodies A and B have a relative motion ; then their 

 velocities towards the centre of inertia will be reciprocally 

 as their masses ; and the momenta in opposite directions 

 will be A.B and B.A. Now if the centre of inertia have 

 also a motion C with respect to a quiescent space, in 

 the direction of A, the velocities will be C;-f B and C — A 

 respectively, and the joint energies will be A.(C-f-B)'-t- 

 B.(C— A)'. But after collision, the velocities B and A 

 relative to the centre of inertia are in a contrary direction^ 

 the motion of that centre remaining the same (28g), there- 

 fore the velocities are C— B and C+A respectively, and the 

 energies A.(C- B)'+B.(C+A)*; but A.(C4-B)'— A.(C— 

 B)"=:2ABC=:B.(C+A)*— B.(C— A)S and the two sums 

 •re equal. 



Scholium. The energy must be estimated in the re- 

 spective directions of the velocities before and after coUi- 



sioa, while the sum of the momenta, which also remaint 

 unaltered, requires to be rcductd to the same direction. 

 The reason of this difference is, that the square of a nega- 

 tive quantity is the same as that of the same quantity taken 

 positively. 



SECTION XI. OF RQTATORT POWER. 



349. Theorem. When a system of bo- 

 dies has a rotatory motion round any centre, 

 the effect of each body in turning the system 

 round a given point must be estimated by 

 the product of its momentum into the dis- 

 tance of tlie body from that point ; and the 

 power of each body, with respect to the ori- 

 ginal centre of rotation, will be expressed by 

 the product of the mass into the square of 

 the distance. 



Suppose the bodies A and B, fixed to the ends of two 

 equal levers, to meet each other, and simply to communi- 

 cate their motion, and let B be twice A, and moving with 

 half its velocity, then the motion of A will exactly destroy 

 the motion of B, and this effect is therefore the measure of 

 the motion of A : but if the bodies A and B be connected 

 with the arms of an inflexible line, and move vvith equal 

 velocities in the same direction, they will be totally stopped 

 by the application of a fulcrum at the centre of gravity ; for 

 the propositions respecting equilibrium are as well deduci- 

 ble from the computation of motion as from that of force, 

 and the motion of A is here equivalent to the motion of B, 

 moving with equal velocity at half the distance : but it was 

 before shown to be equal to the motion of B with half the 

 velocity at its own distance : therefore these two motions of 

 Bare equivalent with respect to effect in producirj; rotatory 

 motion ; and the same may be shown in other cases. And 

 the distance from the centre of rotation being as the velo- 

 city, the power is as the square of the velocity. 



350. Definition. The centre of gyra- 

 tion is a point into which if all the particles 

 of a revolving body were condensed, it would 

 retain the same degree of rotatory power. 



351. Theorem. The centre of gyration 

 of two equal points is at the distance of the 

 square root of half the sum of the squares of 

 the separate distances from the axis. 



