Plots for Hamilton's Variations of the Paths

  In his paper on the Paths of Light and the Planets Hamilton was trying to show what happen to the length of the path as one varies it using circular arcs and elliptic arcs. In each case the minimum length was the length of the straight line when ε=0 and the purpose of the "transformed" expressions was to show that lengths for the nonzero ε's were greater. For both sets of arcs -1≤ε≤1 with ε=±1 corresponding to a semi-circular arc. For ε<0 the arcs are below the straight line. The plots are roughly parabolic in shape but there doesn't appear to be a power series expansion about ε=0 for the set of elliptic arcs. The integral used for the arc length is related to the complete elliptic integral of the second kind. One just has to make the substitution cos²φ=1-sin²φ and one gets k²=1-ε² which indicates that k in the elliptic integral is the eccentricity of the ellipse. Hamilton may have used the set of ellipses since Jacobi had published a work on elliptic integrals a few years earlier.