Finally, he considers the more general case where the speed is an arbitrary function X(x), so the time to be minimised is ( x + a ) X . The minimum condition then becomes X = ( x + a ) d X d x which he writes as : X = ( x + a ) Δ Δ x and which gives MN (=x) as a function of NK (= a). From this the equation of the curve could be obtained from the integral calculus, though he does not demonstrate this.