AAA 2 V r is here = 1-F ; that of the 4 V 8 V third that of the fourth and 1-Fr' 1-Fr3 n-1 so on to the last, wilidiwMe,.— V.
l+r The momentum of the first will be A V, that of the second 2r that that of the third 1r AV that of the fourth 8 r3 AV , and 1 4- r' so on to the last, which will be 1 + r A V.
To give an example of this theorem : if n 100, and r 2, then will the first body A be the last rn-i A, as 1 to 33825300000000000000000000000, nearly; and its velotity to that of the last nearly as 271022000000000000 to 1: lastly, the momentum of the first to that of the last will be nearly as 1 to 2338480000000.
If the number (n) of bodies be required, and the ratio of the moments of the first and last be given as 1 to M, and the ratio of the series r given also; then, putting i R, we have the momentum of the last body expressed by — 1+ r M = therefore the logarithm of M (1. M) is equal to the logarithm of R R) multiplied by the power n-1 ; that is 1. 74=n X 1.R; consequently, 1 = n, the number of bodies re.
quired.
Morton, in botany, implies not so much a change of place, as a change of direction. The direction of the roots and stems of the plants is totally opposite, the former either running directly down wards, or extending themselves trans versely or horizontally under the surface of the earth : the latter exhibiting mo tions of a similar nature, but in a contra• ry direction. The direction of the root is never vertical, except in the sanar of Se negal, the roots of which twisting, rise vertically upwards a foot above the sur face of the earth, and are sometimes co vered by the flux of the sea. Familiar as the appearance is, naturalists are not agreed with respect to the causes which determine the roots of plants to tend uni versally downwards, either in a horizontal or perpendicular direction; and the stems, on the contrary, to mount perpendicular ly or horizontally upwards. So constant, however, are these opposite directions, that a plant being taken out of the earth, and replanted in it in such a manner that the root is uppermost, and the stem low ermost, the root will quickly curve downwards, the stem upwards, till each has resumed the direction which is pro per and natural to itself.
All the causes which concur in promot ing the growth of plants appear likewise to operate in determining their direction. Such are, the air, the sun, light, and the moist warm vapours which arise out of the earth. The three first seem to soncur most certainly to the direction of the stem ; air and moisture to that of the root. If any number of plants are placed in pots, in a room which only admits the light by a single hole, the stems will incline or direct themselves towards that side. In
thick forests, the young trees always lean to the side where the light penetrates. The new shoots of an espalier detach themselves from the wall which robs them of the air, the sun, and the light. It is in quest of the same excellent gifts of nature, that the lateral branches of trees, abandoning the direction of the stem, spread and extend themselves in a direc tion parallel to the soil, even when plant ed on a declivity. In like manner it ap pears, that the roots penetrate more or less deeply into the ground, either in a perpendicular or horizontal direction, in proportion to their greater or less tenden cy to search for moisture. Thus it is a well known fact, that, in the neighbour hood of canals, ditches filled with water, and, ground newly tilled, the roots of plants abandon their natural direction, and as it were, steer their course towards the fine air, rich juices, and grateful hu. midity, which their situation has placed within their reach. So strongly, indeed, are the roots of plants attracted by water, that they frequently relinquish the soil, and penetrate into the very heart of the ditches and canals. The force of exten sion appears to be greater in roots than in stems. The branch surmounts an ob stacle by leaving its natural direction and over-topping it. The root, on the con trary, without once going out of its way, pierces the hardest soils, penetrates into walls, which it overturns, and even into rocks, which it Although the natural motion of the trunk be to ascend, as was suggested above, yet is it forced sometimes to de scend ; for the trunk-roots growing out of some plants near the ground, and shrinking into it, serve, like so many ropes, to pluck the trunk annually lower and lower into the ground along with them. If these trunk•roots break out only about the bottom of the trunk, then it gradually descends into the earth, and is converted into a root, but if the trunk is very slender, and the trunk-roots break forth all along it, then it creeps horizon tally : the trunk-roots in question tether ing it, as it trails along, to the ground, as in strawberry, cinquefoil, and mint It may be observed, that the direction of the roots and stems of plants seems to be regulated, in a great measure, by the va pours which they contain, but more by those which arise from the soil in which they grow ; and that heat, the sun, or the light, the causes already suggested, appear to contribute to that direction, only in so far as they augment or regulate the current of these nourishing vapours.