jSIATHEMATICAL ELEMENTS 



OF 



NATURAL PHILOSOPHY. 



PART III. 



HYDRODYNAMICS. 



OF THE MOTIONS OF FLUIDS. 



SECTION J. OF HYDROSTATIC E^WiLlBR^TJM. 



367. Definition. A fluid is a collection 

 of particles considered as infinitely small 

 spheres, moving freely on each other without 

 friction. 



Scholium. Some have defined a fluid, as a substance 

 which communicates pressure equally in all directions ; 

 but this appears clearly to be a property derivable from a 

 simpler assumption, although, from the deficiency of our 

 analysis, all attempts to investigate mathematically the af- 

 fections of fluids, have hitherto been so unsuccessful, that 

 even this fundamental law can scarcely be strictly demon- 

 strated. 



368. Theorem. The surface of a gra- 

 vitating fluid at rest, is horizontal. 



Suppose two minute straight 

 tubes differently inclined to 

 the horizon, and joined at the 

 bottom by a curved portion, 

 and let them be filled with evanescent spherules : then the 

 relative force of gravity is inversely as the length, when the 

 height is the same ("255), and the number of particles is di- 

 rectly as the length : consequently the absolute pressures 

 will be equal, and there will be an equilibiium ; and if the 

 fluid in cither arm be higher, it will preponderate. The 

 pressure on the tube at any part is only the effect of the pat- 

 VOL. II. 



tide immediately in contact with it, and is communicated 

 in the direction perpendicular to the tube, therefore if ano- 

 ther similar row of particles in equilibrium were placed on 

 the first, this pressure, acting in the same direction, wouJd 

 not disturb the equilibrium of the particles among them- 

 selves, however they might be situated with respect to the 

 first. And conceiving any fluid to be divided into an infi- 

 nite number of tubes, bent or straight, in which the par- 

 ticles form a continuous series, there can be no force to 

 preserve the equilibrium in each of them, unless the height 

 of each portion be equal. Yet some may perhaps hesitate 

 to admit the conclusiveness of this reasoning, without an 

 appeal to our experience of the phenomenon as observed in 

 nature : it may however be admitted by such as an illustra- 

 tion of that phenomenon. 



Scholium. In the equilibrium of fluids, there is some 

 analogy to the general law of mechanical equilibrium 

 (313) ; thus, supposing the whole body of the fluid to 

 begin to move either way, the initial momenta of the par- 

 ticles in the surfaces of the unequal portions of a bent tu&e 

 will be equal. For instance, if one surface be ten times as 

 large as the other, its subsidence will raise the other ten 

 times as much as it sinks. 



369. Theorem. The surface of a gravi- 

 tating fluid, revolving round an axis, is para- 

 bolic. 



The centrifugal force is simply as' the distance from the 

 axis, and may be represented by tlie ordinate, while the 

 I 



