Take a point in the axis for the point about which the moments are measured, and let L be the perpendicular distance from A of the resultant R. Now, as in Art. 106. it appears that the sum of the moments of the effective forces about A is

To determine the value of R let it be observed that the

effective force fm P1 ре on any particle m1, acting in a direction nim1, perpendicular to the distance Am, from the axis A, may be resolved into two others, parallel to the two rectangular axes Ay and Ax, each of which is equal to the product of this effective force, whose direction is n,m,, and the cosine of the inclination of nm, to the corresponding axis. Now the inclination of m1n, to Ax is the same as the inclination of Am, to Ay, since these two last lines are perpendicular to the two former. The cosine of this inclina

AN, tion equals therefore or 1, if AN,=y1. Similarly Am, Pi

the cosine of the inclination of n1m, to Ay equals

AM, or, if Am Pi

AM,=. The resolved parts in the directions of Ay and Ar of the effective force fmp, are therefore fmp p

Y1 and

Similarly the resolved parts in the directions of Ax and Ay of the effective force upon m, are "fmy, and m2, and

so of the rest.

The sums X and Y of the resolved forces in the directions of Ax and Ay respectively (Art. 11.) are therefore

Now let G, and G2 represent the distances GG and G,G of the centre of gravity of the body from Ay and Ax respectively, and let the whole volume of the body be represented by M,

.:. (equation 18), MG2=m1y1+m2Y 2+MzY3+

MG1=m1x1+M2X2 + M3X3 +

... ;

(81).

Now (Art. 11.), R= √X2+Y2, therefore

R="/M√G,+G,
fM

2

2

Now if G be the distance AG of the

centre of gravity from A, G=

2

G ̧2+G ̧2,

Substituting in equation (82) the value of ƒ from equation

And substituting in equation (80) for R its value from equation (82),

where L is the distance of the point of application of the

resultant of the effective forces, from the axis.

Now let be the inclination of the resultant R to the

... tan 0=tan. AGG1, ...0=AGG1.

The inclination of the resultant R to Ax is therefore equal to the angle AGG,, but the perpendicular to AG is evidently inclined to Ar at this same angle. Therefore the direction of the resultant R is perpendicular to the line AG, drawn from the axis to the centre of gravity. Moreover its magnitude and the distance of its point of application from A have been before determined by equations (83) and (84).

THE CENTRE OF PERCUSSION.

109. It is evident, that if at a point of the body through which the resultant of the effective forces upon it passes, there be opposed an obstacle to its motion, then there will be produced upon that obstacle the same effect as though the whole of the effective forces were collected in that point, and made to act there upon the obstacle, so that the whole of these forces will take effect upon the obstacle, and there will be

no effect of these forces produced elsewhere, and therefore no repercussion upon the axis. It is for this reason that the point O in the resultant, where it cuts the line AG drawn from the axis to the centre of gravity, is called the CENTRE OF PERCUSSION. Its distance L from A is determined by the equation

which is obtained from equation (84) by writing MK2 for I (Art. 80.), K being the radius of gyration. If at the centre of percussion the body receive an impulse when at rest, then since the resultant of the effective forces thereby produced will have its direction through the point where the impulse is communicated, it follows that the whole impulse will take effect in the production of those effective forces, and no portion be expended on the axis.

THE CENTRE OF OSCILLATION.

110. It has been shown (Art. 98.) that in the simple pendulum, supposed to be a single exceedingly small element of matter suspended by a thread without weight, the time of each oscillation is dependent upon the length of this thread, or the distance of the suspended element from the axis about which it oscillates. If therefore we imagine a number of such elements to be thus suspended at different distances from the same axis, and if we suppose them, after having been at first united into a continuous body, placed in an inclined position, all to be released at once from this union with one another, and allowed to oscillate freely, it is manifest that their oscillations will all be performed in different times. Now let all these elements again be conceived united in one oscillating All being then compelled to perform these oscillations in the same time, whilst all tend to perform them in different times, the motions of some are manifestly retarded by their connexion with the rest, and those of others accelerated, the former being those which lie near to the axis, and the others those more remote; so that between the two there must be some point in the body where the elements cease to be retarded and begin to be accelerated, and where therefore they are neither accelerated nor retarded by their connexion with the rest; an element there performing its oscillations precisely in the same time as it would do, if it were not connected with the rest, but suspended freely from the axis by a thread

mass.

without weight. This point in the body, at the distance of which from the axis a single particle, suspended freely, would perform its oscillations precisely in the same time that the body does, is called the CENTRE of oscillation.

The centre of oscillation coincides with the centre of
percussion.

111. For (by equation 79) the increment of angular velo-
city per second ƒ of a body revolving about an horizontal
axis, the forces impressed upon it being the weights of
MG
I

its parts only, is represented by the formula 9 sin. 9,

where is the inclination to the vertical of the line AG, drawn from the axis to its centre of gravity. But (by equa

I

tion 84), L= where L is the distance AO of the centre MG

of percussion from the axis,

f= g sin. @

L

..ƒL=g sin. 9.

(86),

Now it has been shown (Art. 98.), that the impressed moving force on a particle whose weight is w, suspended from a thread without weight, inclined to the vertical at an angle 6, is represented by w sin. ; moreover if f' represent the increment of velocity per second on this particle, then is the effective force upon it. Therefore by D'Alembert's principle,

w

9

Now fL is the increment of velocity at the centre of percussion, and f' is that upon a single particle suspended freely at any distance from the axis. If such a particle were therefore suspended at a distance from the axis equal to that