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PROPOSITION XXXV. PROBLEM If a rare medium nitudes, medium from A than any other circular solid whatsoever, VII. consist of very small quiescent particles of equal mag at equal distances from one another = to and freely disposed this jind the resistance of a globe moving uniformly forward in medium. CASE 1. Let a be cylinder described with the same diameter and altitude conceived to go forward with the same velocity in the direction of its axis through the same medium and let us suppose that the particles of the ; medium, on which the globe or cylinder falls, fly back with as great a force of reflexion as possible. Then since the resistance of the globe (by the last Proposition) is but half the resistance of the cylinder, and since the globe is to the cylinder as 2 to 3, and since the cylinder by falling perpendicu larly on the particles, and reflecting them with the utmost force, commu nicates to them a velocity double to its own; it follows that the cylinder. in moving forward uniformly half the length a motion to the density of the medium of its axis, will communicate the whole motion of the cylinder as and that the to the density of the cylinder which the particles globe, in the time it describes is to ; one length of its diameter in moving uni formly forward, will communicate the same motion to the particles and in the time that it describes tw o thirds of its diameter, will communicate ; r a motion is to the whole motion of the globe as the Arid therefore the the density of the globe. a which with the force meets which to its whole mo is resistance, globe by tion may be either taken away or generated in the time in which it de to the particles density of the scribes two sity of the

CASE 2. medium which to thirds of its diameter medium moving uniformly forward, as the den to the density of the globe. Let us suppose that the particles of the medium incident on the globe or cylinder are not reflected and then the cylinder falling per pendicularly on the particles will communicate its own simple velocity to ; them, and therefore meets a resistance but half so great as in the former and the globe also meets with a resistance but half so great. case, CASE 3. Let us suppose the particles of the medium to fly back from the globe with a force which is neither the greatest, nor yet none at all, but with a certain mean force then the resistance of the globe will be in the ; same mean ratio in the second. between the resistance in the first case and the resistance Q.E.I. COR. 1. Hence if the globe and the particles are infinitely hard, and destitute of all elastic force, and therefore of all force of reflexion thf resistance of the globe will be to the force by which its whole motion may ;330 THE MATHEMATICAL PRINCIPLES I) [BOOK be destroyed or generated, in the time that the globe describes four third parts of its diameter, as the density of the medium to the density of the ^lobe. Con. The 2. resistance of the globe, cceteris paribus, is in the duplicate ratio of the velocity. CUR. The 3. resistance of the globe, cocteris paribus, is in the duplicate ratio of the diameter. COR. 4. The resistance of the globe medium. is, cceteris paribus, as the density of the COR, The 5. resistance of the globe is compounded of the du in a ratio plicate ratio of the velocity, arid the duplicate ratio of the diameter, the ratio of the density of the medium. COR. and The motion of the globe and its re be thus Let be the may expounded time in which the globe may, by its resistance uniformly continued, lose its whole motion. 6. AB sistance BC BC Erect AD, be perpendicular to AB. J ,et that whole motion, and through the point C, the asymptotes being AD, AB, describe the hyperbola AB EF Erect the perpendicular CF. Produce to any point E. meeting the the hyperbola in F. Complete parallelogram CBEG, and draw in H. Then if the globe in any time BE, with its first mo meeting AF BC BC uniformly continued, describes in a non-resisting medium the space area of the the expounded by parallelogram, the same in a resisting medium will describe the space expounded by the area of the hv- and its motion at the end of that time will be expounded by EF, perbola; tion CBEG CBEF the ordinate of the hyperbola, there being lost of its motion the part FG. its resistance at the end of the same time will be expounded by the And length BH, there being lost of its resistance the part appear by Cor. 1 and Prop. 3, V., Book CH. All these things II. T by the resistance R uniformly globe in the time whole motion M, the same globe in the time t in a resisting medium, wherein the resistance R decreases in a duplicate COR. 7. Hence continued lose if the its ratio of the velocity, will lose out of its motion M /M the part ,.i the TM part rn . ; remaining scribed in the the number number ; and will describe a space which same time T+ t, is to the space de with the uniform motion M, as the logarithm of t ^. multiplied by the number 2,302585092994 ^ because the hyperbolic area BCFE in that proportion. is to is the rectangle to the BCGEOF NATURAL PHILOSOPHY. SEC. VII.] 331 SCHOLIUM. I have exhibited in this Proposition the resistance and retardation of spherical projectiles in mediums that are not continued, and shewn that this resistance is to the force by which the whole motion of the globe may be destroyed or produced in the time in which the globe can describe two thirds of its diameter, with a velocity uniformly continued, as the density of the medium to the density of the globe, if so be the globe and the particles of the medium be perfectly elastic, and are endued with the utmost force of and that this force, where the globe and particles of the medium reflexion are infinitely hard and void of any reflecting force, is diminished one half. But in continued mediums, as water, hot oil, and quicksilver, the globe as it passes through them does not immediately strike against all the parti ; cles of the fluid that generate the resistance the particles that lie next to press other particles, and so on minished one other half. with a resistance that is ; made to it, but presses only it, which press the particles beyond, which and in these mediums the resistance is di A globe in these extremely fluid mediums meets to the force by which its whole motion may be destroyed or generated in the time wherein it can describe, with that mo tion uniformly continued, eight third parts of its diameter, as the density This I shall endeavour to shew of the medium to the density of the globe. in what follows.