But, a being the are or, we have At the point In question yea, from which the velocities in the directions of x and y are found to be -I- s/(1) and VS!, or and 1.852. We take the positive signs, sae both motions obviously tend to Increaae• their co-ordinates. Differentiate the last equations again, and we have,— or the velocity in the direction of xis always retarded, while that in the direction of y is always accelerated. And at the point in question we have — 4./0-i-49 and 4+49 for the accelerations, say --200 and .032 ; by which we mean that if the pressures then acting In the directions of x and y were allowed to continue uniform for one seirond, they would alter the velocities in the direction of x and y from .756 and p852 to •756—•200 and 1'552+•082. The weight of the particle, if weight • were allowed to act, being 10 ounces, the pressures which would produce the preceding accelerations are, In ounces— the pressures in the direction of x being in the direction from N towards o. At the point in question these presssures are —'06'2 and •0255 ounces.
The pressures thus derived from the motion which actually takes place, by means of the accelerations rtkr : dig and dry : di:, are usually called the effectire forces ; and the name is very appropriate, because it is true that these must be the forces which do really act. Different pressures produce different accelerations upon the same mass ; or to one acceleration there is but one producing pressure, the magi being given. But it may happen that the forces actually impressed. or the pressures actually employed, at the point P. may be very different from those which just produce the motion that is produced. Suppose fon example that the mass P were attached to the masa si by the rigid rod PQ without weight ; and suppose such forces to be applied at r and Q as, whatever may become of ia, cause r to move uniformly along the parabola in the manner above described. We may assign an infinite number of different motions to Q, and fur each motion of Q we may assign an infinite number of pressures which, being applied to r and Q, Will give the two their supposed motions. But in no one of these
cases can the total amounts of pressure really applied to r, in the directions of x and y, be any other than those which are calcu lated above : whatever may be the pressures actually applied at r, the thrust or pull, as the case may be, of the rod PC1, will supply what is necessary to make all the forces that act on r (those directly applied and that arising from the said thrust or pull) together equivalent to the pressures above calculated. This is the foundation of D'Alembert's principle. [FORCES, IMPRESSED AND EFFECTIVE; VIRTUAL VELOCITIES.] The connection between velocity and pressure is not only obscur by phrases as cloudy as "moving force," but also by the use of the unit of mass instead of the unit of weight. This measurement by masses instead of weights is so convenient and so desirable on rational grounds, that it cannot ultimately be dispensed with ; but at the first outset the student should be taught to reduce the new mode of proceeding into terms of that i'vith which he is then better acquainted. A beginner in the theory of gravitation is not allowed to have the least idea of the amount of the attractions of the several bodies upon each other in pounds or tons, or any other unit which he can at once understand. And we should not be surprised if many who can easily compare the sun's attraction upon the earth with the earth's attraction upon the moon, so as to find either of them when the other is given, would be awkward at, if not actually puzzled by, the question of comparing either of them with the weights in a grocer's shop. Undoubtedly there would not be much of astronomical utility iu the question ; but for clear conception of the meaning and mode of derivation of mechanical results, nothing can be of more importance than the actual comparison of all results with those which are best known, because actually felt and perceived.