Page:Somerville Mechanism of the heavens.djvu/84

8 mo, by the simultaneous action of these two forces. For, since the forces are proportional to the spaces, if a be the space described in one second, at will be the space described in t seconds; hence if at be equal to the space mA, and bt equal to the space mB, we have t=$mA⁄a$=$mB⁄b$; whence mA=$a⁄b$mB which is the equation to a straight line mo, passing through o, the origin of the co-ordinates. If the co-ordinates be rectangular, $a⁄b$ is the tangent of the angle moA, for mB=oA, and oAm is a right angle; hence oA:Am::1:tan Aom; whence mA=oA&times;tan Aom=mB.tan Aom. As this relation is the same for every point of the straight line mo, it is called its equation. Now since forces are proportional to the velocities they generate in equal times, mA, mB are proportional to the forces, and may be taken to represent them. The forces mA, mB are called component or partial forces, and mo is called the resulting force. The resulting force being that which, taken in a contrary direction, will keep the component forces in equilibrio.

27. Thus the resulting force is represented in magnitude and direction by the diagonal of a parallelogram, whose sides are mA, mB the partial ones.

28. Since the diagonal cm, fig. 6, is the resultant of the two forces mA, mB, whatever may be the angle they make with each other, so, conversely these two forces may be used in place of the single force mc. But mc may be resolved into any two forces whatever which form the sides of a parallelogram of which it is the diagonal; it may, therefore, be resolved into two forces ma, mb, which are at right angles to each other. Hence it is always possible to resolve a force mc into two others which are parallel to two rectangular axes ox, oy, situate in the same plane with the force; by drawing through m the lines ma, mb, respectively, parallel to ox, oy, and completing the parallelogram macb.

29. If from any point C, fig. 7, of the direction of a resulting force mC, perpendiculars CD, CE, be drawn on the directions of the