The connection between these two subjects is made in the manner described in the article FORCE, according to which it appears that if the weight of a particle be w, and if (this weight being taken away, as by laying the particle on a table without friction) a pressure be con stantly exerted upon it such as would be produced by a weight P, in any direction in which it can move freely, the amount added to the velocity will be uniform, at the rate of 32'19 r w feet in every second. Hence the following equation : 32'19 r dv w da w = Or r = 32'19 dt For example, what pressure must act uniformly for one second on a particle of 7 ounces weight, to add 13 feet per second to its velocity, or that the rate of motion at the end of that second may be 13 feet per second greater than at the commencement ? Here dv : d t =13, w=7, and the answer is 7 x 13 r32.19 ounces.
The numerical divisor 32'19, the uniform acceleration of bodies falling free in vacuo at the earth's surface, is usually denoted by g, and the factor w.÷g usually stands for the Mess of the body, or the measure of its quantity of matter. Hence the following equation : dv r = • lit and this remains true, whatever unit of mass be employed, provided only that the pressure which is called unity shall be that which, exerted for one second upon the unit of mass, shall add a unit to the velocity. [ValtiArtox.] And now comes another consequence of the application of the term force to the simple consequence of force, acce leration. The word is wanted again to signify this pressure which produces acceleration, and for distinction the pressure is called moving force. [MOMENTUM; 310VINO FORCE.] So, then, the name of the prefigure which acts and produces continual accessions to the speed is moving force, while the name of the rate of acceleration is accelerating force. To mend this confused use of terms, some writers endeavour to create a notion of moving force independent of the pressure ; but as they always cud in saying that the moving.force varies as the pressure and never tell us more of its definition than that it is the product of the mass and acceleration, they might save themselves trouble, and their readers also, if they would simply establish the above equation, where r means the pressure which produces acceleration, and that pressure is the unit of its kind when it is of that magnitude which creates in the unit of mass a unit of velocity per second.
As we are now differing from men of deservedly good authority, both at home and abroad, and intending to make our assertion in a more positive manner than is usual with us, we may be excused for dwelling a little more upon the subject. If we consider the natural meaning of moving force and accelerating force, it is obviously as follows :—Moving force is force which makes motion ; accelerating force is force whieh makes acceleration, or increased motion. Were the distinction ever so necessary, these words would be very bad ones, and would always obstruct the learner. Nor does this origin of the word moving force—namely, that which produces MOMENTUM—give any help ; for the synonyme for momentum—namely, quantity of motion, meaning really quantity of matter moved multiplied by the velocity— is a perversion of words of the same kind. To momentum we have no objection ; it is a Latin word to which an English ear may easily be familiarised in any sense. If geometers had chosen to call an equi lateral triangle a momentum, the etymological student might have been startled, but the shock would soon have been got over ; but if they had called the same figure a quantity of every beginner would have been puzzled, and the impression would have been lasting. But returning to the two species of forces, so called, we find a double inconsistency : the idea of motion is introduced into the word which only means pressure (for moving force is but pressure), while the idea of pressure is introduced into a word which has only reference to motion (for accelerating force is but acceleration). There are two
distinct and leading ideas in mechanics, pressure and motion : on keeping them perfectly distinct till the time comes for joining them experimentally it must depend whether a student sees mechanics to be a science or not. if any one should say that pressure producing motion ought to be distinguished from pressure which is in a state of equilibrium with other pressures, we could not of course raise any objection : let, then, moving force and resting force be used in these two senses, with a clearly expressed distinction. Here force would be synonymous with PRESSURE, in the derived or secondary sense of the article cited. But let acceleration be then acceleration, not accelerating force.
The Comrosirrox of velocities and accelerations is so easily proved, that we do not think it necessary to lengthen this article by dwelling upon It. Two of a sort, whether velocities or accelerations, acting upon one particle, at any one instant, are equivalent to a third repre sented in magnitude and direction by the diagonal of a parallelogram, the two sides of which represent in magnitude and direction the two components. And by the law of motion which is commonly called the second (11oTioN, Laws op], the several accelerations which act on any particle is any sirers directions may have their effects com puted separately without any error being introduced. If. then, sup posing a particle to move in a plane, the pressures P and ia be applied to it in the directions of the rectangular co-ordinates x and y, the mass of the particle being at, we have equations which are only true on the supposition that there is this connection between the unit of mass and that of pressure, namely, that the latter acting on the former during one unit of time shall add to the line which represents its velocity one unit of length. These orations are enough to determine the equation of the curve In which the particle must move, P and Q being given functions of both x and y; and the time of motion through any arc of the curve a is then found from the following equation :— It is not here our business to proceed further with the consequences of the definitions of velocity and acceleration; but we must explain a point which will arise in our subsequent article on VIRTUAL VELO CITIES. When we have the means of actually ascertaining the motion of a particle of given mass,—that is to say, of finding at every instant its actual place, its velocities in the directions of its co-ordinates, and its accelerations in those directions,—we are prepared to assign the pressures which must act upon it in those directions, at the instant we are speaking of, either in mathematical unite of pressure, as before described, or, If the reader please, in pounds or ounces avcrdupois. To show this, let us propose an instance, as follows :—A particle whose weight (if weight were allowed to act) is 10 ounces, moves uniformly along the are of a parabola o r (whose focus is 8, o s being half a foot) at the rate of 2 feet per second : What pressures in the directions of ON and N P (or of r and y) are necessary to keep up the motion ; and in particular what are the pressures and the velocities at the point r at which N r = 3 feet / The equation of the curve is 2y = e, whence we get dy dr x c77 Or the velooity in the direction of y is to that in the direction of x always as x to 1.