A being impelled by both thofe forces or impulfes, at the fame time, will move in a direction AC, between AD and AB; for fince, according to the fecond law of motion, the change of motion is always made in the direction of the right line in which that force is impreffed, therefore the motion of the body along the line AD is altered from the direction A D, to another direction towards AB, by the other impulfe, which acts in the direction AB. And for the fame reason, the motion of the body along the line A B is changed for another direction towards AD by the impulfe which acts in that direction. Therefore the motion arifing from thofe two impulfes must have a direction between AD and A B.-But it will be fhewn in the following propofition, how much this new direction will deviate from AD, and from A B.

II. When a body is impelled at the fame time by two forces in different directions, if two lines be drawn from the place in which the body receives the double impulfe, in the directions of thofe impulfes; and the lengths of thofe lines be made proportionate to the impelling forces; alfo through the end of each of thofe lines a line be drawn parallel to the other, a parallelogram will thereby be formed; and if a diagonal line be drawn from the place where the body receives the double impulfe to the oppofite corner of the parallelogram, the length and fituation of that diagonal will reprefent the velocity and the direction of the body's motion, arifing from the double impulfe.

Thus, fuppofe that the body A, fig. 26, Plate II. be impelled in the direction A D, by a force which would enable it to move at the rate of 4 feet per fecond; also that at the fame time the fame body be impelled by another force in the direction AB, which would enable it to move at the rate of 3 feet per fecond. Make AD equal to four, and AB equal to three (for inftance, inches; or you may use any other dimenfion to reprefent feet). Through D draw DC parallel to A B; and through B draw BC parallel to AD; by which means the parallelogram A B D C will be formed. Laftly, draw the diagonal AC, and A C is the direction in which the body which is impelled by the above-mentioned two impulfes, will move. Alfo the length A C will exprefs the velocity of the body; fo that if AC be found, either by calculation or by measuring the diagram, to be 5 inches long (1); we conclude

that

(1) The length and direction of A C; viz. the angles it makes with AD and AB, may be eafily found by trigonometry; it being the folution of a plane triangle, in which two fides, and the angle between those two fides, are known.

The direction of the impulfes being given, the angle DAB is also known; for it is the angle which the directions of the two impulfes make with each other. The angle ADC is likewise known, because it is the complement of the angle DAB to two right angles. The lines AD and DC (AB) are to each other in the pro

portion

that the body will move at the rate of 5 feet per fecond, fince in the dimenfions of AB and AD, inches were employed for reprefenting feet.

That the body thus impelled by the two forces must move along the line A C, is eafily deduced from the second law of motion; for fince the change of motion is proportionate to the moving force impreffed; if from any point in the diagonal AC you draw two right lines, viz. de parallel to A B, and be parallel AD, thofe two lines will represent the deviations of the body's motion from the directions AD and AB; fince by law the 2d, the change of motion is made in the direction of the moving force impreffed. And those two lines are proportional to the impelling forces, portion of the two impulfes, and may be reprefented by any dimenfions, as inches, feet, &c, Therefore in the triangle ADC the fides AD, DC, and the included angle D, are known. Hence by trigonometry we have AD + DAC + DCA

DC: ADDC:: tangent
DCA - DAC

2

2

:: tangent

; whence we obtain half the fum of the

angles at the base, viz. of the angles DCA and DAC. Then half the fum, plus half the difference, is equal to the greater of those angles; viz. DCA; and half the fum, minus half the difference, is equal to the other angle DAC, which gives the direction of AC: thus all the angles will be known. Laftly, fay, as the fine of the angle DCA is to the fine of the angle ADC, fo is AD to a fourth proportional, which is equal to AC. Hence both the direction and the length of AC will be known.

or to the lines A B and A D, which reprefent; those forces; viz. de is to be, as AB is to AD; because (by Eucl. p. 24. B. vi.) the parallelogram Adcb is fimilar to the parallelogram ABDC.

If it be faid that the body thus impelled will at any time be found at any other place out of the diagonal AC, draw om parallel to A D, and od parallel to AB; then om and od, which represent the deviations, &c. ought to be proportional to the forces which occasion those deviations, viz, om qught to bear the fame proportion to ed as AD does to AB. But this is not the cafe, because the parallelogram Admo is not fimilar to the parallelogram ADBC. Therefore the body, &c. muft move along the diagonal AC, and in no other direction, III. When a body is impelled at the fame time by three forces in three different directions, the velocity and the direction of the body's motion, which arifes therefrom, must be determined by first ascertaining the courfe which would be produced by any two of thofe forces, according to the preceding propofition; and then by finding the course laft found, and the third force, which will be the courfe fought..

Thus if a body A, fig. 27, Plate II. be impelled by three forces; viz. with a force which by itself would enable it to move in the direction A B at the rate of four feet per minute; by a fecond force, which by itself would enable it to move in the direction AC at the rate of three feet per minute; and laftly, by a third force, which by itfelf would enable it to move at the rate of five feet per mi

nute

nute in the direction A D. Make the lengths of the right lines proportionate to the forces, viz. A B four, AC three, and AD five, inches, or feet, &c. long. Then imagine as if the body were impelled by the first and second forces only, and, by the preceding propofition, find the compound motion arifing therefrom, viz. through B draw BE parallel to A C, and draw CE through C parallel to AB; and the diagonal A E will reprefent the direction and the velocity of the motion refulting from thofe two forces. Then after the fame manner find the compound motion refulting from the force reprefented by A E, and the third force represented by AD; viz. by drawing through E and D the lines EF and DF, respectively parallel to AD and to A E, as alfo the diagonal A F; and this diagonal A F will represent the course of the body, viz, the velocity and direction of its motion, arifing from the above-mentioned three impulfes.

The demonftration of this propofition is fo evident a consequence of the preceding propofition, that it will be needlefs to detain the reader with a repetition of almoft entirely the fame words.

It appears likewife, that the like reasoning may be extended to the cafe of four or five, or in short, of any number of impulfes..

Notwithstanding the apparent multiplicity and intricacy of such cases, an obvious remark will furnith a general rule, by means of which the place of the body at any time may be easily determined

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