on the Cohesion of Fluids. 6g 
conjointly, or as the ordinate inversely, and directly as the 
sine of elevation ; therefore the fluxion of the ordinate multi- 
plied 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 radius, is equal to 
the fluxion of the versed sine ; therefore the ordinate multi- 
plied by its fluxion is equal to the initial height multiplied by 
the fluxion of the versed sine in the corresponding circle of 
curvature ; and the square of the ordinate is equal to the rectangle 
contained by the initial height and twice 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 con- 
stant quantity is therefore equal to the rectangle contained by 
the initial height and its difference from twice the diameter : 
this constant quantity is the square of the least ordinate , and the 
ordinate is every where a mean proportional between the greatest 
height and the same height diminished by twice the versed sine of the 
angular depression in the corresponding circle of curvature. Again, 
at the vertical point, the square of the ordinate is equal to the square 
of the greatest height diminished by the rectangle contained by this 
height and the diameter of the corresponding circle of curvature , a 
rectangle which is constant for every fluid, and which may be 
called the appropriate rectangle : deducting this rectangle from the 
square of the ordinate at the vertical point , we have the least ordinate; 
which consequently vanishes when the square of the ordinate at 
the vertical point is equal to the appropriate rectangle; the horizontal 
surface becoming in this case an asymptote to the. curve , and, thee 
