68 
Dr. Young's Essay 
and taking the reciprocal of the remainder. In this case the 
analysis leads to fluxional equations of the second order, which 
appear to afford no solution by means hitherto discovered ; 
but the cases of simple curvature may be more easily subjected 
to calculation. 
III. Analysis of the simplest Forms. 
Supposing the curve to be described with an equable angular 
velocity, its fluxion, being directly as the radius of curvature, 
will be inversely as the ordinate, and the rectangle contained 
by the ordinate and the fluxion of the curve will be a 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 
the horizon ; and the fluxion of the area varying as the cosine, 
the area itself will vary as the sine of this angle, and will he 
equal to the rectangle contained by the initial ordinate , and the 
sine corresponding to each point of the curve in the initial circle 
of curvature. Hence it follows, first, that the whole area in- 
cluded by the ordinates zvhere the curve is vertical and where it is 
horizontal, is equal to the rectangle contained by the ordinate and 
the radius of curvature ; and, secondly, that the area on the 
convex side of the curve, between the vertical tangent and the 
least ordinate, is equal to the whole area on the concave side 
of the curve between the same tangent and the greatest 
ordinate. 
In order to find the ordinate corresponding to a given 
angular 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 as the radius 
of curvature and the sine of the angle formed with- the horizon 
