But if two of the nearest of the points assumed in the arches of the curves, on the same side of the axis, be supposed to accede to one another, beyond what.
ever limits, and at last to coincide, which will be done by making two M equal, and likewise two N equal, then the curve sought will touch the arch of the given curve there, and if three such points coin cide, it will osculate it ; nay, as many points as we please may be made to meet together• where we please, and thus we may have osculations of what or der we please, and as near one another as we please, the arch of the given curve approaching as we please, and at whatever distances we please, to whatever arches of whatever curves, and yet -still preserving all the six conditions required for expressing that law of the repulsive and attractive forces. And whereas the value of T can be varied in infinite manners, the same may be done in an infina•umber of ways, and there fore•a simple curve, answering the given conditions, may be found out in an number of ways. Q. .E. F.
It would be very possible to divide this curve, though single in itself, into two or more. Thus, if -any one should wish to consider. general gravity as accurately the• squares of the distances, he may describe on the attractive side the hyperbola •which expresses it•vhich, in fact will be a continua tion of the leg VTS, then to each of the ordinates lid, of the curve of forces, Fig. 1. he may add the ordinates of hyperbola towards the parts AB, beginning at the points g.and h of the curve. By which means a new curve will arise, coinciding to sense with the axis, towards the parts V p, but else where at a distance from it, and•even_winding about it, if the vertices F, K, 0, arc more distant than the hy perbola is. In this way even inany different forces may be expressed, which, as in the resolution of forces, may sometimes be found useful in more - readily de monstrating the And, in fact, it will be a true of forces : but, nevertheless, it is merely the offspring of the mind: As we have mentioned the decrease of gravity to be accurately in the reciprocal duplicate ratio of the distance, which is generally, admitted by the culti vators of mechanical astronomy,' it may seem an ob jection against our theory, that it departs so widely from that law. But, in the first place, the action of
the particles in the lesser distances does very much .differ from that law, seeing that vapours, which exert such a great force of expansion, must have, in these distances, a repulsion to each other, and, not an at traction, and that even the attraction of cohesion is vastly greater than that which should be produced by general gravity ; and hence some of the disciples of Newton have supposed a force corresponding to 1 this formula 13 .z2 —, the former part of which is im .r mensely less than the latter• -when s is greater than the assumed unity, but is greater than it when is much less ; so that in the greater distances the force is -very nearly inversely as •the squares, and in the less very nearly as the cubes of the distances. So' :that the duplicate ratio is not strictly adhered to.even among the Newtonians.
It has indeed been demonstrated by Newton, that the line of apsides of the planets would have an im mense motion, if. the ratio of forces were very distant from being inversely as the squares pf the distances. But they have some motion, and it is not enough to say that this is owing to the disturbing force of the other planets, for this is not yet accurately demon strated; and indeed it is only after many attempts and approximations, that a partial solution has been gi ven of the celebrated problem of three bodies, in which there is sought the motion of three bodies act ing-on each other in the inverse duplicate ratio of the distances. It may be guessed, therefore, how far we yet are from having any demonstration of the strict accuracy of this law of forces.
To many the greatest difficulty in the theory ap pears tb be the total rejection of immediate contact, which, they think, is evidently shown by the testi mony of the senses ; a rod, they say, should be used -to him who denies contact. But it is admitted, that bodies approach so near each other as to leave no sen sible distance between them ; and that the resistance we experience is produced by the repulsive power, which gives us the same sensation as actual contact -is supposed to do : the contact being physical, al though not mathematical.