La Place next examines how for these results would he changed, if other relations subsisted between the force and the velocity. Force, he observes, may be expressed in a great variety of was relatively to the velocity, besides that of the simple law of proportionality, without implying any mathematical contradiction. Suppose the tierce to be some other function of the velocity, (analytically expressed by F = V j) in this case the principle of the vis viva will be fiend to obtain in all the possible mathematical relations between the f-ree and the velocity ; the vis viva of a body being the product of its mass by double the integral of its velocity. multiplied by the differential of the function of the velocity indicative of the force.
The uniform rectilinear motion of the centre of gravity is preserved by the law of nature alone ; by which also the conservation of areas subsists. But a principle analogous to that of the least action will be found to belong to every possible relation between force and velocity.
The principle of the least action is not so obvious as the others that have been mentioned, being more remote from the elementary theorems, from which they are all derived ; nevertheless, if it be directly and mathematically deducible from the same simple principles, it must immediately he divested of all pretension to the dignity of a final eause ; to which it can ha% e no greater claim than any other remarkable numerical property ; for the rever-c would imply a mathematical contradiction, The flirt. which is curious, may be this analytically stated : Suppose a material point to move under the impulse of several forces from one point to another ; the curve which it describes possesses the remarkable property of haying the integral or continued product of the velocity (determined by pre ions considera tions) when multiplied into the element of the curve, less than any other eurve passing through the same points.
Maupertuis, who discovered this principle, carried it no further than to single bodies; Eulerestablished itsgenerality ; and La Grange extended it to a system of bodies acting on each other, as already noticed. The principle of the least action being therefore admitted as an established theorem, it may be resorted to for the solution of problems, and for determining the trajectories of bodies moving in space ; but in point of practical utility, the necessary calculations are so much more complex and difficult than the more usual methods of investigation, that the latter are greatly to he preferred to it.
Upon these principles of the 'doctrine of forces, as laid clown by the most eminent writers on the subject, it is to be remarked, that they are for the most part mere develop ments of theorems easily deducible from the Newtonian laws of motion ; and that many of them were even established by Newton himself. This generalization of mechanical prin
ciples possesses, however, the advantage of enabling us to take a more enlarged and comprehensive view of the subject, than we could do by the consideration of a single problem.
Forces, which become the subject or mathematical compu tation. may be appropriately divided into the three following classes : 1. Such as act instantaneously, or for a -bort, interval of time, and hnpart uniform motion to a particle subjected to their action ; provided it be not solicited by any force, and is free to move in any direction.
2. Such as. acting with a continued uniform intensity, oblige a material particle, at liberty to obey their hnpulse, to describe its path with a uniformly accelerated motion.
3. Those, whose intensities. perpetually varying, though according to some known law, produce a complicated action, whose circumstances can only be investigated by means of the integral calculus, or some analogous methods.
Forces whose mode of action is too arbitrary and uncertain to be included in either of these classes, may be considered as foreign to the present investigation.
The reader who wishes to have more scientific informa tion on this subject, may refer to Gregory's Mechanics ; Dr. Jackson's Theoretical Mechanics ; and may also con sult Marat's Wood's Mechanic's, or Whewell's Mechanics. Professor Leslie also. (in his Elements of Natural Philosophy,) has given an excellent popular illustration, by supposing the threads to act on light spiral springs adapted to measure the forces,and commonly called spring steel-yards; but he acknowledges pulleys and weights have some advan tages. By reversing the action of the springs, they might be applied, with much advantage, to show the relations of compressing forces. by lecturers on mechanical science.
FORCE, Desagtdiers, in his Experimental Ph ilosophy. has many curious and useful observations con cerning the comparative forces of men and horses, and the best mode of applying them. And Dr. Young. in his Lec tures, has given a table of a similar nature, compiled chiefly from the writings of Desaguliers and Coulomb, another writer, who has also displayed considerable ingenuity in pursuing the subject. The Ibilowing extract cannot ftil of being use ful to all concerned in practical mechanics.