OF ROTATOny POWER. 



5.1 



The distance of the points from the axis being a and b, 

 the whole rotatory power will be «*+//, which is equal 

 to the sum of the particles multiplied by the square of 



352. Theorem. The distance of the cen- 

 tre of gyration of a right line from an axis 

 at its extremity, is to its length, as 1 to v^3. 



The fluxion of the rotatory power is x''i, consequently 

 the whole rotatory power is ^r', which is equivalent to the 

 effect of .T at the distance of i/lx'. But if the centre of mo- 

 tion does not coincide with the end of the line, the rotatory 

 power will be the sum or difference of the two values of .r 

 at the end of the line, as \{a^±b'], and the distance of the 

 centre of gyration becomes •/ (^(o'd:''')), divided by a±l. 



353. Theorem. The distsince of the 

 centre of gyration of a circle or any circular 

 sector from its centre of rotation and of cur- 

 vature is to the radius as 1 to v'Q. 



The area of any increment of the circle, of which the 

 radius is x, will be as x*',and its rotatory power x*!', the flux- 

 Ion x'x and the fluent ^r' ; but the whole area will be as 

 fr', and the rotatory power the same as if the whole were at 

 the distance v' (i^*)- 



354. Definition. The centre of percus- 

 sion is a point in which an obstacle must be 

 placed in order to receive the whole effect of 

 the motion of a revolving body, without pro- 

 ducing any pressure on the axis. 



335. Definition. The centre of oscil- 

 lation of a body is a point of which the dis- 

 tance from the axis of motion is equal to the 

 length of a pendulum vibrating in tlie same 

 time with the body, 



356. Theorem. The centres of percus- 

 sion and of oscillation coincide always in 

 the same point. 



The effect of the velocity of every part of the body, re- 

 duced to the direction in which the obstacle opposes it, is 

 expressed by the product of each particle, into its distance 

 from the line drawn through the axis, parallel to that direc- 

 tion : now the joint effect of all these reduced momenta is 

 equal to the resistance of the obstacle, since the axis is sup 

 posed to be free from any pressure in consequence of the 

 percussion ; and the resistance of the obstacle acting at the 

 given diitancc is also equivalent to the rotatory power of the 



whole body. But the sura of the reduced momenta is als* 

 expressed by the product of the whole mass into the distance 

 of the centre of gravity from the line drawn through the 

 axis (27fl) which is equal, acting at the distance of the 

 centre of percussion, to the whole rotatory power, or the 

 sum of the products of all the particles by the squares of 

 their distances, and the distance of the centre of percussion 

 from the centre of suspension is found by dividing the 

 rotatory power by the mass and the distance of the centre 

 of gravity. 



In the same mannerj when a body is suspended as a pen- 

 dulum, the tendency of the weight of each particle, to turn 

 it round the axis, is proportional to the distance frorh the 

 vertical line passing through the point of suspension ; and 

 the sum of the forces of all the particles is expressed by the 

 product of the whole weight into tlie distance of the centre 

 ofgravity from the same line; and the rotatory mass to be 

 moved is to be estimated by the joint products of the'parti- 

 cles into the squares of their distances: and in order that 

 the angular velocity of the equivalent pendulum may be 

 equal, its distance from the vertical line must be to the 

 square of its distance from the centre, in the same ratio, as 

 the product of the distance of the centre of gravity into the 

 whole weight,to the rotatory mass ; but the distance of these 

 points from the vertical line is as the distance from the cen- 

 tre, therefore the distance of the centre of oscillation is ex- 

 pressed by the rotatory mass divided by the weight and the 

 distance of the centre of gravity from the point of suspen- 

 sion ; consequently it is equal to the distance of the centre 

 of percussion from the same point. 



Scholium. It may also be shown that the distance of 

 the centre of oscillation from the centre of gravity varies in- 

 versely as the distance of the centre of suspension from the 

 same point. 



357. Theorem. The centre of oscilla- 

 tion of two equal points in a right line pass- 

 ing through the'axis is found by dividing 

 the sum of the squares of iheir distances by 

 the sum or diflerence of their distances. 



For the rotatory power is a'+i'', and the weight mul» 

 tiplied by the distance of the centre of gravity is a±/'. 



S58. Theorem. The centre of oscilla- 

 tion of a right line suspended at its extre- 

 mity is at the distance of two thirds of its 

 length, 



7 he fluxion of the rotatoiy power is x'i, the fluent ^', 

 the distance of the centre of gravity ^, the product ix', and 

 the quotient j*. 



