the force ƒ acting in the direction LC, can only be a resolved part Let these two parts of the resolved force be denominated Y' and X', let TM = x', and ML = y, then, in order to determine the body's motion, we shall have two equations similar to the two preceding, namely, Two lines as TM, ML, perpendicular to each other and serving to determine the place of the body L, are technically called, Rectangular Co-ordinates of the body L. Instead of them, we may use two others, such as TK, KL (x, y) and obtain two equations similar to the preceding. For, X' can be resolved into two other forces in the directions of TK, KL, and Y' also may be resolved into two forces in the same directions. Hence, if X, Y, represent the resulting forces in the directions of a and y, we As any force tending from L towards T may be resolved into two forces X, Y, acting in directions parallel to x, y, so may any other force F', and any number of forces F", F", &c. be resolved entirely into partial forces acting in the same directions. Hence, if X, instead of representing the resolved part of a single force F, should represent the result of the several resolved parts of the forces, F, F', F", &c. in a direction parallel to x, and Y the result of the forces parallel to y, the two preceding equations We may give to these equations a different form, by substituting instead of X and Y, their values expressed in terms of the entire forces, F, F', &c. For instance, suppose the body to be solicited by a single force F, then, if r = √(x2 + y2), or, since, according to the differential notation, (See Analyt. Calc. p. 79.) F, in the above equations, represents the centripetal force tending towards T. The equations, it is plain, cannot be solved except we assign to F a specific value; and such value will depend on what is called the Law of the Variation of the force. Suppose the law to be that of the inverse square of the distance of the body L from T; then, assuming to be a determinate quantity, we may represent Fby, and the two preceding equations will be Hitherto we have taken account of only one plane, namely, that in which the co-ordinates x, y are situated, and in which the forces X, Y, act. And, it is not essentially necessary to consider more than one plane, so long as the forces, whatever they be, continue to act in it. It might, however, even in these circumstances, be convenient, to consider the body's position and motion relatively to a second plane. For instance, the forces that act on Mercury lie almost entirely in the plane of the orbit of that planet. If they did so exactly, still we should find it convenient to introduce, inclined to the orbit's plane, a plane like that of the ecliptic, to which Astronomers are accustomed to refer the positions and motions of heavenly bodies. But if the forces do not lie all in the same plane, then it becomes absolutely necessary to take account of more planes than one. As two rectangular co-ordinates x, y, are sufficient to determine the distance of the body L from T in the plane of LK, KT, (see Fig. p. 4.) so, three x, y, z, (z being drawn perpendicular to the plane of x, and y) will determine the distance when the body is either above, or below, the plane in which x and y are. And, whatever be their directions, the forces that act on a body, or on a system of bodies, may all be resolved into three directions respectively parallel to the co-ordinates x, y, z. For, let TK be x, K1, perpendicular to it, y, and let LI perpendicular, at the point /, to the plane passing through x, y, be z; then, if the force acting on L were represented by any line drawn obliquely to the plane of x and y, it might be resolved into two others, one parallel to L (z), the other lying in the plane of x and y and this may be effected by merely drawing from one extremity of the line representing the force, a line parallel to z, and, from the other extremity, a line parallel to the plane of x and y, and produced to the former line. The force represented by the second line, lying either in the plane of x and y, or in a parallel plane, being then, by a second process of resolution, resolved into two directions parallel to x and y respectively, the whole force would be resolved into three others parallel to x, y and z. In the figure, it is plain that TI=√(x2 + ́y3), and TL = √(x2 + y2 + z3). Tl, if TL be called the radius, is said to be the projected radius. In order therefore to determine the body's motion, &c. when the forces do not lie in the same plane, we must introduce a third equation similar to the two that have been already introduced. The three differential equations of motion will then be in which, as in the former case (see p. 6.), we may substitute The body's place has been supposed to be determined by means of three rectangular co-ordinates x, y, z; and, there is no other more simple way of determining it. But, if we look to the custom of Astronomers, this is not the usual mode of determining it. A body's place (see Astronomy, p. 252.) is made to depend on the length of the radius vector (or on that of the curtate distance or projected radius) and on its latitude and longitude. It is made therefore to depend on, one line and two angles, instead of, three lines. |