
42, Geometrical Theory of a Plane Non-Cyclic Are, 
Finite as well as Infinitesimal. 
By Prof. Syamapas Moxnopapuyaya. 
InTRopUCTION. 
manner. An analytical curve is one which is represented by an 
analytical equation. In an analytical curve, the curvature 
and its rates of variation, oh Se ete., of all possible orders, 
s 3? 
are necessurily finite and continuous, except at a certain limited 
number of points. In a geometrical curve, no such restriction 
necessarily hol e may, however, study such geometrica 
curves by supposing that the curvature and its rates of variation, 
us. 
he following paper is an attempt to study geometrically a 
plane arc, under the supposition that the radius of curvature 
only is finite and ec mtinuous, or that the radius of curva. ure, as 
well as its first rate of variation, is finite and continuous. No 
ted 
methods have been suggested, and a number of interesting 
In the first place, consecutive points, on a curve, have been 
defined as the intersection of the curve with a line of given species 
&, these consecutive points being only the position of ultimate 
coincidence, of a number of real distinct points, which must have 
originally existed in every Case, separated by finite distances. The 
one. In counting consecutive 
points, the analytist, not infrequently, confounds real points with 
imaginary ones. The point of undulation is an instance. 
i 
is really more fundamental than the complete rate of variation, 
only it does not come go naturally in the analytical way, 
attempted by introd 
which might stand independently. 
