DYNAMICS. 



lionatc parallelogram, fig. 5. 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, arc 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 compute 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- 

 lerating force ; 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- 



nerated in unequal bodies are also con- 

 jointly as tlte forces, and their times 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 in any times are directly 

 as the forces and times ; and reciprocal- 

 ly as the bodies or musses. 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 : 01 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 ot 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 eflort 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- 

 lustration, 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 DB, then the 



