COHESION OF FLUIDS. 17^ 



laf velocity, its fluxion, being directly as the radius of cur- On the simplest 

 vaturc, will be inversely as the ordinate, and the rectangle con- g'^^^ce^of a ^ 

 tallied by the ordinate and the fluxion of the curve w ill be a, fluid. j, 



constant quantity ; but this rectangle is to the fluxion of the 

 area, as the radius to the cosine of the angle formed by the 

 curve with its fluxion ; and the fluxion of the area varying, 

 as the cosine, the area itself will vary as the sine of this angle,, 

 and mil be equal to the rectangle contained by the initial ordi- 

 nate, and the sine corresponding to each point of the curve in' 

 the initial circle of curvature. Hence it follows fast, that the 

 xvhole area included by the or dinates laherethe curve is vertical 

 and "where it is horizontal, is equal to the rectangle containsd 

 by the ordinate and the radius of curvature; and,.secondh> 

 that the area on the convex side of the curve, between the 

 vertical tangent and the least ordinate, is equal to the vvhola 

 area on the concave side of the curve between the same tan- 

 gent and the greatest ordinate. 



In order to fn>d the ordinate corresponding to a given an- 

 gular direction, we must consider that the fluxion of the 

 .ordinate at the vertical part, is equal to the fluxion of the 

 circle of curvature there, that, in other places, it varies 'a% 

 the radius of curvature and the sine of the angle formed with 

 the horizon conjointly, or as tliiC ordinate inversely, and di- 

 rectly as the sine of elevation ; therefore the fluxion of the 

 ordinate multiplied by the ordinate is equal to the fluxion of 

 any circle of curvature multiplied by its corresponding height, 

 and by the sine, and divided by the radius : but the fluxion of the 

 circle multiplied by the sine and divided by the tadiusis equal 

 to the fluxion of the versed sine; therefore the ordinate multiplied 

 by its fluxion is equal to the initial height multiplied by the 

 fluxion of the versed sine in the corresponding circle of cm-- 

 vature ; and the square of the ordinate is equal to the rectan- 

 gle contained by the initial height and txvice the versed sine, 

 increased by a constant quantity. Now at the highest point 

 of the curve, the versed sine becomes equal to the diameter, 

 and the square of the initial height to the rectangle contained 

 by the initial height and twice the diameter, with the constant 

 quantity : the constant quantity is therefore equal to the 

 rectangle contained by the initial height and its difference 

 from twice he diameter: this constant quantity is the square •'■'■'■" 



of 



