IANICAT, r II I LOSOPHY. HYDROSTATICS. 



ATTR< 



th normal pressure "f I' mi the surface H P D ; so that 

 this normal pressure ia the resultant of the other two, 

 ami therefore, taken in the opposite direction P N, 



ratines the former, and keeps V at rest 

 Lot a be the angular 4 rotation, that is, tho 



angle turned through byMP or FC in a second of time; 

 then tin- i-irriilar arc turned through by 1' in a second 

 will l>o ay, puttin;.; v fur P H ; that i to say, the Telocity 

 ,i- point 1', ia v ay. 

 the centrifugal force acting on P is 



^-^-o'y. (Di-SAMics, page 741). 



y y 



p. IV., page C9G), 

 N M : M I' : : gravity : centrifugal force 



(1) 



that ia, N M : y : : g : oy .'. NM- 



The line NM, thus determined, is called the sub- 

 normal to the curve A H D at the point P ; and we see 

 that the curve is such that the subnormal at any point 

 is a constant quantity. This property belongs exclusively 

 to the paraMa. Hence the surface of equilibrium is 

 that generated by the rotation of a parabola about its 

 axis ; that is, it is a paraboloid. 



In reference to the foregoing investigation, it may 

 be useful to the student to attend to the following 

 remarks : 



It will have been perceived that the problem proposed 

 is of a mixed character partly dynamical and partly 

 statical. The forces first generate motion in P, which 

 point continues to move till these forces become balanced, 

 and P takes a position of rest. The force of gravity g, 

 and tho centrifugal force n*y, both act dynamically, and 

 are therefore to be expressed in feet. The symbol a, re- 

 presenting the angular velocity of rotation, is an abstract 

 number it is not an angle. Here, as in all dynamical 

 inquiries, angular velocity is estimated thus : A circle, 

 whose radius is represented by 1, is imagined to be 

 described about the centre of motion (about M in the 

 present case) ; the semicircumference of this circle is 

 3 '1416 ; and the extent of arc of tho same circle, turned 

 through in a second, is a, the angular velocity of the 

 rotation, which is therefore 3 '1410 multiplied by some 

 number, whole or fractional. If, therefore, y or M P be 

 measured in inches, the velocity of P is ay inches ; if in 

 feet, it is ay feet. As g is measured in feet viz. g= 32 '2 

 feet, it is necessary that M P or y should also be 

 measured in feet ; otherwise tho proportion (1) would be 



incongruous. The final result, namely, NM=^, gives 



the length of N II in feet : it is 32-2 feet divided by the 



abstract number a*. Suppose the vessel performed 10 



rotations in a second ; then the angular velocity would be 



1-3-1416 X 20=62-832 .'. a j = 62 -832 s =3947 -80 



We see that, with the same angular velocity of rotation, 

 the curve is the same whatever be the magnitude of the 

 vessel : it is further obvious that, as the cylindrical 

 sliape of the vessel is no item of consideration in the 

 above solution, the conclusion remains the same what- 

 ever the shape may be, provided that the horizontal 

 section* be circles, and the vertical axis of revolution 

 pas* through their centres. 



Suppose the vessel to be itself in the form of a para- 

 boloid : then the velocity of rotation, necessary to cause 

 all the fluid to run over, may be easily determined. For 

 the vowel will be emptied as soon as every point P has, 

 through the intensity of the centrifugal force, reached 

 the side of the vessel ; that is, when the subnormal N M 

 at any j-.int 1' is identical with the subnormal at the 

 point where the horizontal line M P meets the side of 

 HMTI -i 



Culling, then, the constant subnormal of the given 



parabola I, wo bare J_ .-. _ 



, the angular ve- 



locity per iecond at which the vowel must rotate in order 



that the empty paraboloid, formed by the rotation, may 



it the concavity of the vessel itself. 

 From this expression for the angular velocity, the 

 time in which each complete revolution must be 

 formed may be ascertained thus : 



a:31410::lsec. : 3 ' U1 . sec, -time of a semi-rotation 



a 



.'.time of arotation-2il 4 l^-3-1416x2 x / X - 1 seconds 

 a 9 



the subnormal I being measured in feet, and g being I 

 32-2 feet. It appears, therefore, that with this'ra- 

 rotation, the fluid will rise and run over the vessel, till 

 only what covers and adheres to the sides is left. 

 The following figure will illustrate tho manner of coin. 

 Fig. IM 



mnnicating rapid rotatory motion to a vessel of w 

 D and E are two pulleys, round which passes an endless 

 cord ; by means of a winch C, the vessel may be made 

 to rotate round its vertical axis A B as rapidly as we 



CAPILLARY ATTRACTION AND REPULSION. 

 Common observation shows that the surfaces of solids in 

 general, unless they are unctuous surfaces, attract tho 

 particles of water : this is more especially noticeable in 

 glass. If a plate of glass be dipped in water, ami then 

 gently withdrawn, a line of fluid particles will be 

 suspended from the lower edge, and the whole surface 

 will be wetted, showing that there exists an attraction 

 between glass and water. 



It is to this attraction that the rise of water in a capil- 

 lary tube is attributed : the bore of such a tube being 

 very minute, from ^th to j^th o f an j ni:n in diameter, the 

 thread of water is so slender that the surface of glass, 

 above its upper extremity, may exercise attractive force 

 sufficient to raise the upper particle, when the contiguous 

 particles must follow to fill up the vacuum : this lengthen- 

 ing of the thread increasing till gravity, or the weight of 

 the little column, counterbalances this capillary attrac- 

 tion. It is observed that the height to which the water 

 is raised in a capillary tube, is inversely as the dian 

 of the tube. And this fact is sufficient to teach us the 

 form of the curve which the upper surface of water 

 assumes, when acted upon by the two glass plates which 

 cinlirncu it 



For the distance between the plates at B (Fig. 184), 

 Fi(r 1M is to the distance between them 



at C, as the height of the liquid 

 at C, to the height at B. Hence, 

 putting AB x, and AC=x', 

 and the corresponding heights y 

 and y', nud remembering, from 

 Euclid, Prop. II., Hook VI., that 

 tin' distances between the ]> 

 at I', ami are as A B to AC, 

 we have 



x : x' : : y' : i/ .'. ry-x'y'; 

 that is, the rectangle xy is a 

 constant quantity, wherever B may !>. within the limits 

 of capillarity. As this is the distinguishing property of 

 the hyperbola, the axes of reference being the asymptotes, 



