Dividing by R, multiplying by cos. ", and transposing

Now sin. (2a)-sin.

)}=2 sin. (a—1) cos. a.

sin. {a+(a)}—sin. {a-(a

Substituting this value of 2 sin. (a) cos. a in the preceding equation, we have

Now it is evident that if a be made to vary, remaining the same, R will attain its greatest value when sin. (2a-) is greatest, that is when it equals unity, or when 2a

2'

or when a This, then, is the angle of elevation

corresponding to the greatest range, with a given velocity upon an inclined plane whose inclination is .

「 ྋ

If in the preceding expression for the range we substitute

—(a—1) ¦ for a, the value of the expression will be found

to remain the same as it was before; for sin. (2a) will, by this substitution, become sin. -2(a-)-sin. -(2x -)=sin. (2x-). The value of R remains therefore the

same, whether the angle of elevation be a or

2—(a−1). And the projectile will range the same distance on the plane, whether it be projected at one of these angles of elevation or the other.

Let BAC be the inclination of the plane on which the projectile ranges, and AT the direction of projection. Take DAS equal to BAT.

T

2

Then BAT-TAC-BAC And SAC-DAC-DAS=

is therefore the same, whether TAC or SAC be the angle of

elevation, and therefore whether AT or AS be the direction of projection.

Draw AE bisecting the angle BAD, then the angle EAC

=BAC+BAE=BAC++BAD=+1(−1) = +2.

4

The angle EAC is therefore that corresponding to the greatest range, and AE is the direction in which a body should be projected to range the greatest distance on the inclined plane AB.

It is evident that the directions of projection AS and AT, which correspond to equal ranges, are equally inclined to the direction AE corresponding to the greatest range.

120. The velocity of a projectile at different points of its path. It has been shown (Art. 56.), that if a body move in any curve acted upon by gravity, the work accumulated or lost is the same as would be accumulated or lost, provided the body, instead of moving in a curve, had moved in the direction of gravity through a space equal to the vertical projection of its curvilinear path.

Thus a projectile moving from A to P will accumulate or lose a quantity of work, which is equal to that which it would accumulate or lose, had it moved vertically from M to P, or from P to M, PM being the projection of its path on the direction of gravity. Now the work thus accumulated or lost equals one half the difference between the vires vivæ at the commencement and termination of the motion.

Let V equal the velocity at A, and v equal the velocity at

P, therefore the work V2.

W
g

W

v'. Moreover, the work

g

W.

W

done through PM=W. PM, therefore V2 — §

W. PM, therefore V'-v'=2gMP. Let PM=y,

which determines the velocity at any point of the curve.

CENTRIFUGAL FORCE.

121. Let a body of small dimensions move in any curvi

B

linear path AB, impelled continually towards a given point S (called a centre of force) by a given force, whose amount, when the body has reached the point P in its path, is represented by F. Let PQ be an exceedingly small portion of the path of the body, and conceive the force F to remain constant and parallel to itself, whilst this portion of its path is being described. Then, if PR be a tangent at P, and QR be drawn parallel to SP, PR is the space which the body would have traversed in the time of describing PQ, if it had moved with its velocity of projection from P alone, and had not been attracted towards S, and RQ or PT (QT being drawn parallel to RP) is the space through which it would have fallen by its attraction towards S alone, or if it had not been projected at all from P. Let v represent the velocity which it would have acquired on this last supposition, when it reached the point T. Therefore (Art. 66.), if w represent the weight of the body,

Now the velocity v, which the body would have acquired in falling through the distance PT by the action of the constant force -F, is equal to double that which would cause it to de scribe the same distance uniformly in the same time.‡ Representing therefore by V the actual velocity of the body in its path at P, we have

Substituting this value of v in the preceding equation,

The force here spoken of, and represented by F, is the moving force, or pressure on the body (see Art. 92.), and is therefore equal to that pressure which would just sustain its attraction towards S.

+ See Art. 113. (equations 89 and 90); what is proved there of a body acted upon by the force of gravity which is constant, and whose direction is constantly parallel to itself, is evidently true of any other constant force similarly retaining a direction parallel to itself. To apply the same demonstration to any such case, we have only indeed to assume g to represent another number than 324.

If represent the additional velocity per second which F would communicate to the body, and t the time of describing PT, then (Art. 44.)

v=ft; but (Art. 46.) PT==(); so that is the velocity with which PT would be described uniformly in the time t.

FxPT=22 V2 (PT),

..F=2y, QR. 9 (PR) Now let a circle PQV be described having a common tan gent with the curve AB in the point P, and passing through the point Q. Produce PS to intersect the circumference of this circle in V, and join QV; then are the triangles PQV and QPR similar, for the angle RQP is equal to the angle QPV (QR and VP being parallel), and the angle QPR is equal to the angle QVP in the alternate segment of the cirQR PQ. cle. Therefore PQ PV; therefore QR=

=

(PQ)2

PV

Substi

tuting this value of QR in the last equation, we have

F=2

Now this is true, however much PQ may be diminished. Let it be infinitely diminished, the supposed constant amount and parallel direction of F will then coincide with the actual case of a variable amount and inclination of that force, the

PQ

ratio will become a ratio of equality, and the circle PR

PQV will become the circle of curvature at P, and PV that chord of the circle of curvature, which being drawn from P passes through S. Let this chord of the circle of curvature be represented by C,

The force or pressure F thus determined is manifestly exactly equal to that force by which the body tends in its motion continually to fly from the centre S, and may therefore be called its centrifugal force. This term is, however, generally limited in its application to the case of a body revolving in a circle, and to the force with which it tends to recede from the centre of that circle; or if applied to the case of motion in any other curve, then it means the force with which the body tends to recede from the centre of the circle of curvature to its path at the point through which it is, at any time, moving. When the body revolves in a circular path, the circle of curvature to the path at any one point evidently coincides with it throughout, and the chord of curvature becomes one of its diameters. Let the radius of the circle which the body thus describes be represented by R, then C=2R;

Since in whatever curve a body is moving, it may be con ceived at any point of its path to be revolving in the circle of curvature to the curve at that point, the force F, with which it then tends to recede from the centre of the circle of curvature is represented by the above formula, R being taken to represent the radius of curvature at the point of its path through which it is moving.

If a be the angular velocity of the body's revolution about the centre of its circle of curvature, then V=aR;

Now (Art. 94.)

Fa

го

represents the additional velocity per second f, which would be communicated to a body falling towards S, if the body fell freely and the force Fremained constant. Moreover, by Art. 47. it appears, that V is the whole velocity which the body would on this supposition acquire, whilst it fell through a distance equal to C, or to one quarter of the chord of curvature. Thus, then, the velocity of a body revolving in any curve and attracted towards a centre of force is, at any point of that curve, equal to that which it would acquire in falling freely from that point towards the centre of force through one quarter of that chord of curvature which passes through the centre of force, if the force which acted upon it at that point in the curve re mained constant during its descent. It is in this sense that the velocity of a body moving in any curve about a centre of force is said to be THAT DUE TO ONE QUARTER THE CHORD

OF CURVATRE.

123. The centrifugal force of a mass of finite dimensions.

Let BC represent a thin lamina or slice of such a mass contained between two planes exceedingly near to one another, and both perpendicular to a given axis A, about which the mass is made to revolve.