ON THE COHESION OP FLUIDS. 



651 



will be expressed by— ; the fluxion of the 

 absciss will therefore be — , t being the co- 



ry o 



sine of the arc z, and r the radius, and the 

 fluxion of the area will be — But — is the 



r T 



fluxion of the sines of the arc z in the circle 

 to which it belongs; consequently, the area 

 is expressed by as, and is equal to the rect- 

 angle contained by the initial ordinate, and 

 the sine corresponding to each point of the 

 •curve in the initial circle of curvature. Hence 

 it follows, that the whole area (ABEF or EF 

 GH) included by the ordinates where the curve 

 is vertical and where it is horizontal, is equal 

 to the rectangle contained by the ordinate and 

 the radius of curvature. 

 Inordertofindtheordinatey,corresponding 

 to a given angular direction, and to a given 



_ _ , ave +i/= — , or, since — is the 



y=: , and 



fluxion of the versed sine v, 



■:^yy—av, vihence yy=b-:+.9.av. But at the 

 summit of the curve, when v — 0, y=-a, 

 therefore b=:aa, and yy=.aa — Q.av\ and 

 where the curve meets the absciss, y=0 and 

 a=2». lfa=4r, when 3^=0, t) will be 2r, 

 and the curve will touch thehorizontallineat 

 an infinite distance, since its curvature must 

 be infinitely small; if a be greater than 4r, 

 the least ordinate will he\/{{ia — Aar). When 

 the curve is vertical, vzzr, and yy=.aa — 

 2«r. The rectangle, contained by the ele- 

 vation above the. general surface, and the 

 diameter of the circle of curvature, which 

 is here lar, is constant in all circumstances 

 for the same fluid, and may therefore be 

 called the appropriate rectangle of the fluid ; 

 and when the curve is infinite, and a = 4ar, 

 this rectangle is equal to 8rr, or to \^aa, so 



that r and a may be readily found from it: 

 it is also equal to the square of the ordinate 

 at the vertical point, where yy^aa — lar. 

 If we describe a circle (ABCD Plate 15. 

 Fig. 114.) of which the diameter is a, the 

 chord of the arc of this circle (AC, AB,) cor- 

 responding in angular situation to the curve, 

 will be equal to the ordinate (EF, GH,) at 

 the respective point ; for the versed sine in 

 this circle will be 2c, and the chord will be a 

 mean proportional between a and a — 2r ; in 

 this case therefore, where the curve is infi- 

 nite, the ordinate varies as the sine of half 

 the angle of elevation. 



For determining the absciss, it would be 

 necessary to employ an infinite series ; and 

 the most convenient would perhaps be that 

 which is given by Euler for the elastic curve, 

 in the second part of the third volume of the 

 Acta Petropolitana. 



IV. APPLICATION TO THE ELEVATION OF 

 PARTICULAR FLUIDS. 



The simplest phenomena, which aflbrd us 

 data for determining the fundamental pro- 

 perties of the superficial cohesion of fluids, 

 are their elevation and depression between 

 jjlates and in ca[)illary tubes, and their ad- 

 hesion to the surfaces of solids, which are 

 raised, in a horizontal situation, to a certain 

 licight above the general surface of the 

 fluids. When the distance of a pair of 

 plates, or the diameter of a tube, is very mi- 

 nute, the curvature may be considered as 

 uniform, and the appropriate rectangle may 

 readily be deduced from the elevation, recol- 

 lecting that the curvature in a capillary tube 

 is double, and the height therefore twice as 

 great as between two plates. In the case of 



