Let us suppose that H and I, fig. 4. two equal weights, were to be raised by means of a chord from each meeting in K : the line of force which should raise them with equal velocity would be along the diagonal K L. If the weights were unequal, the line would not be the dia- . gonal of an equilateral quadrangle, as in the preceding case, but along the parallel Q V of the diagonal formed by a propor tionate parallelogram, fig. 3. The angle being more acute with the line of the greater weight B, and more obtuse with that of the lesser weight C, the paral lelogram would be formed of sides ' cor responding with the proportions of the two weights; which are in this instance considered at 4 to 1.
When various bodies, moving in the same course, are acted upon equally by a second power, not in a line with that of the primary momentum, they will each preserve their relative situations in regard to each other. We have a familiar proof of this in the descent of rain, &c. which will be found to preserve a parallel course among the drops respectively, though they may each be driven from the line of gravity, and be impelled into an oblique direction, by the force of wind. The case is similar among the vessels of a fleet, supposing all to sail alike, when they get into a current ; they will preserve their relative situa tions, and perform all their evolutions, precisely as though they were in stagnant water.
Uniformly varied motion relates to bo dies, which, at regular periods, or at re gular distances, receive new impulses, either in the same direction, or in oppo. sition thereto ; in the first case, the new momentum is to be added to the former velocity : in the latter case, it is to be deducted from it. In variable motions we the velocity, according either to a medium taken at the average of the accession of forces, or we consi der the velocity at the instant chosen for calculation to be equal to an unit ; i. e. a second of time ; we may thus calculate each unit separately. A force causing a body to move quicker is termed an acce leratingforce ; though the term is also ap plied to forces which cause the motion to vary: when it occasions a regular increase of velocity, by an equable addition of im pulse, it is termed an uniformly accelerat ing force ; but if it occasions a body.to move regularly slower, as an additional weight in mechanics, or a lengthened pen dulum in clock-work, it is called an uni formly retarding force.
A single body would, by a constant force, give four objects of consideration, viz. the space, the time, the velocity, and the force : any three of these being given, the fourth may be found. For the velocities generated in equal bodies, by the action of constant forces, are in the compound ratio of the forces, and the times of acting. The momenta ge iterated in unequal bodies are alaciara jointly as the forces, and their of action ; and the momenta lost or de stroyed in any times are likewise con jointly as the retarding forces, and their times of action : for an equal opposing force would, in equal time, destroy the impelling force. The velocities generat
ed or destroyed ill any times are directly as the forces and times ; and reciprocal ly as the bodies or masses. In all mo tions uniformly accelerated, when the force and body are given, the space de scribed during a certain time is the half of that which the body, moving uniform ly with the last acquired velocity, would describe in an equal time ; and the spaces described by a body uniformly accelerated are as the squares of the times; hence the velocities acquired be ing' as the times, we shall find the spaces described to be the squares of the velo cities. Therefore, either the velocities, or the times, are as the square roots of the whole of the described spaces. And this applies equally to motions uniformly retarded.
Of variable motions in general. By this we immediately refer to those forces which act incessantly, but always in dif. ferent degrees of strength; of this we have an instance in the release of springs from heavy pressure ; whence they gra dually recover the form in which they were made. A carriage set in motion, though the horses may appear to act uni. formly, nevertheless occasions less exer tion to them, when it has acquired its due degree of velocity. For bodies in a state of rest are, in a manner, disposed to remain so; and bodies in motion are disposed, in a certain degree, to continue so, as will be subsequently shewn : there fore we find the first effort to be consi derable; while, on the other hand, it of ten requires much resistance on the part of the horses, even on level ground, to discontinue that motion of which their forces were the cause. Besides this il luAtration, we find that bodies are very sensibly affected by the forms of those spaces over which they have to pass, admitting the ascents to be equal, and the whole to be perfectly level, and free from impediments; which, indeed, must ever be understood, while treating of the motion of bodies, especially by com parison. If a body X should have to pass over the point A, fig. 6, along the plane A B, its motion would be uniform ly continued by the power C ; but if the surface were concave, as A D B, then the first part of the motion would be more rapid than the lattter ; because the com mencement of the concave space being more horizontal than the latter, or upper part, would permit the force to act more powerfully on the body D. On the other hand, if the line of ascent were convex, as at A E B, the first part of the ascent, being steepest, would be slowest ; and the latter, in consequence of its regular approach to the horizontal, would be proportionately rapid. Thus we see that C is an uniform force on the plane A B ; an accelerating force on the con vex ascent A E B ; and a retarding force on the concave ascent A D B: we there fore deduce, that accelerating and retard ing motions may be described by arcs of which the axes are easily ascertained.