of indirect functions will show how to deduce from this formula in each particular case, when the equation of the curve is given, the relation between t and x, by eliminating the auxiliary quantities which have been introduced. If we suppose, in order to complete the solution, that the equation of the proposed curve is y = ax2, we shall evidently have
Δy = 2axΔx + a(Δx)2,
from which we shall obtain
Δy/Δx = 2ax + aΔx.
Now it is clear that the limit towards which the second number tends, in proportion as Δx diminishes, is 2ax. We shall therefore find, by this method, t = 2ax, as we obtained it for the same case by the method of Leibnitz.
- Rectifications. In like manner, when the rectification of a curve is desired, we must substitute for the increment of the arc s the chord of this increment, which evidently has such a connexion with it that the limit of their ratio is unity; and then we find (pursuing in other respects the same plan as with the method of Leibnitz) this general equation of rectifications:
(LΔs/Δx)² = 1 + (LΔy/Δx)²,
or (LΔs/Δx)2 = 1 + (LΔy/Δx)2 + (LΔz/Δx)2,
according as the curve is plane or of double curvature. It will now be necessary, for each particular curve, to pass from this equation to that between the arc and the abscissa, which depends on the transcendental calculus properly so called.