Page:Somerville Mechanism of the heavens.djvu/85

Chap I.] component forces mA, mB, these perpendiculars are reciprocally as the component forces.

That is, CD is to CE as CA to CB, or as their equals mB to mA.

30. Let BQ, fig. 8, be a figure formed by parallel planes seen in perspective, of which mo is the diagonal. If mo represent any force both in direction and intensity, acting on a material point m, it is evident from what has been said, that this force may be resolved into two other forces, mC, mR, because mo is the diagonal of the parallelogram mCoR. Again mC is the diagonal of the parallelogram mQCP, therefore it may be resolved into the two forces mQ, mP; and thus the force mo may be resolved into three forces, mP, mQ, and mR; and as this is independent of the angles of the figure, the force mo may be resolved into three forces at right angles to each other. It appears then, that any force mo may be resolved into three other forces parallel to three rectangular axes given in position: and conversely, three forces mP, mQ, mR, acting on a material point m, the resulting force mo may be obtained by constructing the figure BQ with sides proportional to these forces, and drawing the diagonal mo.

31. Therefore, if the directions and intensities with which any number of forces urge a material point be given, they may be reduced to one single force whose direction and intensity is known. For example, if there were four forces, mA, mB, mC, mD, fig. 9, acting on m, if the resulting force of mA and mB be found, and then that of mC and mD; these four forces would be reduced to two, and by finding the resulting force of these two, the four forces would be reduced to one.

32. Again, this single resulting force may be resolved into three