H Y D R O D Y N A M I C S. 



543 



stance in a direction FG, perpendicular to the plane, 

 is proportional to the square of the sine of the angle of 

 inclination, it must itself vary as the sine of the angle 

 when reduced to the direction DE; that is, the whole re- 

 sistance must vary as sin.* ABC x sin. ABC =; sin. 5 ABC, 

 or the cube of the sine of the angle of inclination. 



SCHOLIUM. 



The two preceding propositions enable us to com- 

 pare together the resistances experienced by plane sur- 

 faces moving in a fluid with different velocities, and at 

 different angles of inclination. It is necessary, how- 

 ever, to know the absolute measure of the resistance for 

 perpendicular impulses, in order that we may determine 

 the absolute measure* in all other cases. It follows 

 from the experiments of Bossut, that the resistance ex- 

 perienced by a plane surface, which strikes directly 

 and perpendicularly an indefinite fluid, is sensibly 

 equal to the weight of a column of the fluid, which has 

 for its base the area of the plane surface, and for its 

 altitude the height due to the velocity of the surface, 

 that is, if R is the resistance, A the area of the resisted 

 surface, * the specific gravity of the fluid, and h the 

 height due to the velocity, we have R=A 2 A. 



If the fluid it not of indefinite extent, but is merely 

 a vein which strikes a plain surface at rest, the abso- 

 lute measure of the resistance is quite different, being 

 equal to a little less than the weight of a column of 

 fluid whose base is the area of the surface, and whose 

 height it double the height which is due to the ve- 

 locity of the i-suing vein, or R=i A2A. This mea- 

 sure of the resistance was first determined accurately 

 V. . Krafft. Duhamel had made experiment* on 

 the same subject in 1669, and several other philo- 

 sophers followed in the same path ; but the result 

 which they obtained was, that the height of the column 

 was equal only to that which was due to the velocity. 

 Kraft employed a rectangular lever, against the verti- 

 cal arm of which the water issuing from an additional 

 tube impinged, while the weights requisite to balance 

 that force were placed in a scale placed on the hori- 

 zontal arm. In this way he obtained the results which 

 we have already given in p. 415. See Comment. PC- 

 tropol. 1736, vol. viii. p. 853. 



M. Bossut has given an account of similar experi- 

 ments in the Sd edition of his Hydrodynamics, vol. 

 ii. p. 293, from which it follows that the resistance is 

 little less than the weight of a column whose height is 

 double that which belongs to the velocity. In Bostut's 

 experiments the water issued vertically from the bot- 

 tom of a reservoir, upon one of the arms of a horizon- 

 tal balance ; and be observed that the resistance was 

 always less when the flat arm of the balance touched 

 the orifice, than when there were some interval be- 

 tween them. 



These two simple propositions constitute the whole of 

 the ordinary theory of the resistance of fluids. They are 

 founded upon two suppositions, neither of which are 

 correct. 1. That after any particle of fluid strikes the 

 plane, it is supposed to be annihilated, or to produce 

 no farther effect ; and, 2. That the part of the force 

 which in oblique impulses is parallel to the surface of 

 the plane, has no influence whatever upon the resist- 



ance which the plane experiences from the perpendicu- Resistance 

 lar part of the force. The first of these suppositions of Fl>"ds. 

 is obviously incorrect, for as the particles are not anni- "*" ""V""' 

 . hilated after impulse, they must some how or other get 

 out of the way of the particles which succeed them, 

 which can only be done by acting upon them, and 

 consequently affecting their velocity. The second 

 supposition appears also to be incorrect, for Mr Vince 

 found that the part of the force which is parallel to the 

 plane is not entirely lost. 



By proceeding upon the principle laid down in the 

 preceding proposition, it is easy to determine the re- 

 sistance experienced by globes, cones, cylinders, and 

 in short bodies of any form. Such determinations, 

 however, are of no use, as they differ too widely from 

 the experimental results to be capable of any practical 

 application. 



For a general account of D'Alembert's theory of the 

 resistance of fluids, we must refer our readers to the 

 historical part of this article, where a short notice of 

 Don George Juan's theory is also given. The expla- 

 nation of this last theory, however, belongs more pro- 

 perly to Mechanics, as it is a deduction from that au. 

 thor's physico-mathematical theory of percussion. 



SECT. II. Account of Experiments on the Resistance of 

 Fluid*. 



* 



I. Account of Bosiul't Experiments. 



IN our History of HYDRODYNAMICS, we have already Account of 

 given a general account of the experiments made by Boaut't ex- 

 Bossut, LXAIembert, and Condorcet, in the years 1775, 

 and by Bossut alone in 1778, on the resistance of fluids. 

 The following were the leading results which they ob- 

 tained. 



1 . That the resistances of the same body of any fi- Heristaneei 

 gure which divides a fluid with different velocities, are > n n '"''' 

 sensibly proportional to the squares of these velocities. finue fluid * 



2. 1 hat the direct and perpendicular resistances of 

 plane surfaces, are sensibly proportional for the same 

 velocity to the area of the surface. 



3. That the resistances which arise from oblique im- 

 pulses are not in the ratio of the squares of the sines of 

 the angles of incidence ; but that, when the angles of 

 incidence are between 50 and 90, the ordinary theory 

 may be employed as an approximation, by observing that 

 it always gives resistances a little less than experiment, 

 and as much less as the angle differs from 90. 



4. That the absolute measure of the direct and per- 

 pendicular resistance of a plane surface in an indefinite 

 fluid, is the weight of a column of the fluid, which has 

 for its base the area of the surface, and for its altitude 

 the height due to the velocity. 



The resistance is much greater, and nearly double, 

 in a mill course, which conveys the water to the float- 

 boards of an undershot wheel. 



5. That the tenacity of the water is a force which 

 may be regarded as infinitely small, in relation to that 

 which a body experiences in striking the water, par- 

 ticularly when the velocity is considerable. 



The next set of experiments by Bossut were made Rruitances 

 on the resistances experienced in narrow canals. Dr in narrow 

 Franklin had ascertained, by a rude experiment, when 

 he was travelling in Holland, that barges experienced 



The experiment* of the Cheraliir Borda gar* m different remit. He employed a cube, which floated upon the itagnant water 

 of a large bain. and be found that tbe height due to the rcsteunce did not much exceed that which wu due to the velocity. Se 

 Mem. jtcud. Ptr. 1763, 1767. 



