258 
Mr. Atwood’s Disquisition on 
if the ordinates are drawn parallel to the axis ; y and x signi- 
fying the ordinate and corresponding abscissa ; the other let- 
ters denoting constant or invariable quantities, to be determined 
by the properties of the figures. In these figures, it is observ- 
able, that the breadths toward the vertex O, are always greater 
in the curves which are of the higher dimensions ; and, as the 
dimensions are continually increased, the figure approaches 
more nearly to a rectangular parallelogram,* with which it 
* The radius of curvature of the conic parabola at the vertex (fig. 23 .) is half the 
principal parameter ; but, in all the parabolas of the higher orders, the radius of cur- 
vature at the vertex is infinite. Suppose x to represent the abscissa, or distance of the 
ordinate y from the vertex, measured along the axis of the curve : as x increases from 
o, the radius of curvature decreases till it becomes a minimum, and then increases : a 
difficulty seems to arise respecting the magnitude and variation of the radius of cur- 
vature, when, the dimensions being increased sine limite, the form of the curve ap- 
proaches continually, and ultimately coincides with, the rectangular parallelogram. If 
the equation of the curve be y n — p n — 1 x x, where/) represents the parameter, the 
radius of curvature of the curve at the extremity of an ordinate y, of which the 
abscissa is x, will be found — p X 
2 71 2 
271 2 
3 
2. 
^ 71 
n x x 
+ P * 
, which quantity 
n — 1 n — 2 * J 
is a minimum when x — p X 
71 
n — 2 
2 71 2 
zn 3 — n x 
w X « — 1 X/> ” xx n 
consequently, the least radius of cur- 
3 n — 3 
vatu re itself, or r — p x 
n — 2 
n — 1 zn 3 — n x 
n % n—z 
271 1 
• ; and, when n is increased sine 
zn - — 2 
limite , the abscissa corresponding to the least radius of curvature, or x — p'x — 
V z xn 
27 
and the least radius itself, or r ~p x — — -, both of which quantities are evanescent, 
shewing that if the dimensions of the parabola are increased sine limite, the curvature 
at the extremity of the ordinate, when the abscissa — o, is infinite, the radius of cur- 
vature being nothing, as it ought to be, at the point H of the parallelogram SHOD, 
