HYDRODYNAMICS. 



415 



>iny. against carved sar&ce*. and it is of no service whatever 



determining the resistance of fluid* when the r t 

 ing body if completely submerged. In order to put 

 this theor * of experience, Daniel Bernoulli, 



ami his pupil IWessor Krafit. instituted a series of ex- 

 periments on the impulse of a stream of water against 

 plain surface placed horizontally. These experi- 

 ments, which are highly valuable, are published in the 

 Mh and 1 Uh volumes of the Commentaries of St Pe- 

 tersburgh. The stream of water was received on a 

 plain surface fixed on the arm of a balance, which had 

 a Male suspended at the opposite extremity. The 

 weights in the scale were made to balance the resist- 

 ance produced by the impulse on the surface, and the ve- 

 locity of the issuing fluid was determined from the dis- 

 tance to which it waa projected on a horizontal plane. 

 These results were wonderfully conformable to the de- 

 ductions of theory. The experimental was always a 

 little less than the theoretical resistance, as appears 

 from the following results. 



Rcctuaacc by theory 



BcnUttoce bj uperinwnt 



1701 

 1403 



I7t0 

 1463 



1651 

 1406 



100* 

 1401 



1 S2O 

 Ito3 



1071 

 10S1 



,>i -. M -- 



Mlli. 



John Bernoulli, the father of Daniel, was occupied 

 with the subject of Hydrodynamics at the same time 

 with his son ; and there i* reason to believe, that so 

 early a* the year 17%, he was in possession of the chief 

 '** part of his theory of running water. The work which 

 he CQBBBOSsd upon thi* subject renamed in MS. till 

 his death, when it appeared in 1 74* in the fourth vo- 

 lume of his works, under the title of lludmlica mrac 

 primtnm dttttta tt dtrecte dtmonttrmta t* 'principal ptat 

 aMMsscu. It wa* also published in the Memoirs of 

 the Acaoemy of St Petenburgh for 1737 and 1738. 

 The method of John Bernoulli is founded upon an as- 

 sumption not very unlike to the Newtonian cataract ; 

 and the principal results of hi* theory did not differ very 

 widely from those of his son. In the opinion of the 

 celebrated La Grange, it is defect* 

 but Euler, who bad seen the MS. 



noulli, in a letter prefixed to the work, for having disco- 

 vered the true prim . drodyrumics. In his dis- 

 course on tne law* ot tne communication off motion, 

 John Bernoulli has determined, on the same supposi- 

 tion, the resistance of fluids ; but the formula, by which 

 he rt presents the resistance, though sufficiently simple, 

 i* still insufficient. 



About the same time, our countryman, Conn Mac- 

 -. r objected to the theory of Daniel Bernoulli, in 

 so far as he employed the doctrine of the conservation 

 mg forces, and endeavoured to solve the problem 

 < motion of fluids that are discharged from reser- 

 t>y a more direct method. This method, which is 

 only an extension of the theory of Newton, was pub- 

 reatise on Fluxions, which appeared in 

 It i- gnrn under two sections, one of which 

 treats of the motion of water issuing from a cylindric 

 vessel, t,d the second of the motion of water 



be method has not been 

 ed as sufficiently rigorous. 



Tbe ci. Ir.vilirt was now destined to re- 



ceive the most important accessions from the genius of 

 the celebrated D'Alembert. When he wa. 



km 1 7 1 T. j " '!"* ** lheof y, f Pendulums given by J-im-, 



Incovercd hi* famous dynamic-it! prin- 

 > termining the motion of a system of solid 

 bodies which act upon each other. He consider* the 

 Telocity with which each body tend* to move, a* com- 



SET 



ir 



M 



posed of two other velocities, one of which is destroy- in^tory. ^ 

 ed, while the other does not obstruct the motion of the """"V^" 

 adjacent bodies. In applying this principle to hydrau- 

 lics, he first enquires what ought to be the motion of 

 the particle* of a fluid, in order that they may not ob- 

 struct one another's movements. He shews, both from 

 theory snd experiment, that when a fluid issues from a 

 vessel, its upper surface always preserves its horizon- 

 tali ty, from which he concludes that the velocity of all 

 the points of any horizontal stratum, when estimated 

 in a vertical direction, is the same, and that this velo- 

 city, which is that of the stratum, ought to be in the 

 inverse ratio of the area of the base ot' the stratum it- 

 self, in order that it may not obstruct the motions of 

 the other strata. By combining this principle with the 

 general one, D'Alembert has reduced all tae problems 

 relative to the motion of fluids to the ordinary laws of 

 hydrostatics. The problems which relate to the pres- 

 sire of fluids against the sides of vessels in which they 

 run, and to the motion of a fluid which escapes from 

 a vessel moveable and carried by a weight, though 

 they had formerly been solved only by indirect me- 

 thod, flow a* corollaries from U'Alembert's general 

 principles. This theory has also the great advantage 

 of enabling us to demonstrate, that the doctrine of the 

 conservation of living forces applies to the motion of 

 fluids as well as to that of solids ; and the principles of 

 the theory are applicable to elastic as well as inci 

 fluid*, and to the determination of the motion of fluids 

 in flexible pipes, a case which applies to the me- 

 chanism of the human frame. These fine views were 

 first published at the end of D'Alembert'a Dynamics in 

 1743, and they were afterwards more fully developed in 

 his Traitf dt Fttfuilibrt et du Mottvemtnt det Flutdrt, 

 which appeared at Paris in 1 " 



After having established the law* of the equilibrium D'Alrmbcrt 

 and motion of fluids, D'Alembert next directed hi* aU <" lhe re- 

 tention to the resistance which they oppose to the mo- '" of 

 tion of solid bodies. This eminent mathematician at- 

 tributes the slow progress of discovery in this branch 

 of hydrodynamics to those unphilosopfaical investiga- 

 tions, in which a greater fondness wa* shewn fur the 

 calculu* than for the physical principles on which it is 

 (bunded ; and he doe* not scruple to say, that the 

 choice of then principles was often made, more from 

 their forming a good ground work for the application 

 of the calculu*, than from their having a real founda- 

 tion in nature. In order to avoid this error, D'Alem- 

 bert fttst investigated the principles upon which he 

 wa* to proceed before he thought of the analysis which 

 bo was to apply to them; and by this truly philo- 

 sophical mod* of enquiry-, be has established a theory 

 founded upon no arbitrary supposition*. He merely 

 auBBSssi that a fluid i* a body comuuml of very small 

 particle- , detached, and capable of moving freely among 

 one another. D'Alembert regard* the resistance 

 which one body suffers from another as nothing more 

 than the quantity of motion which it lose* ; and when 

 the motion of a body i* changed, he considers this mo- 

 tion as composed of thst which the body will have in 

 the following instant, and of another which is destroy- 

 ed Tiii- principle he found applicable to the resist- 

 ance of fluid*, and the investigation of this resistance 

 he reduces to the laws of equilibrium between the fluid 

 and the solid body. He supp*es at first, that a body 

 is by some external mean* kept at re*t in the middle of 

 a (luid which is about to strike it. When the filaments 

 fluid strike the solid, they bend themselves round 

 it in different directions, and that part of the fluid 



