Curves in Hardy's "A course of pure mathematics"

Question

In section 229 of Hardy's "A course of pure mathematics," 10th ed, a curve, $C$, along which an integral is to be defined, is given by $x=\phi(t)$, $y=\psi(t)$. It is said that as $t$ varies from $t_0$ to $t_1$ the point $(x,y)$ moves along C in the same direction. Does that mean that $\phi$ and $\psi$ are monotonic and not both constant?

Answer 1

Pretty sure it just means that we consider the curve as moving from $(\phi(t_0),\psi(t_0))$ to $(\phi(t_1),\psi(t_1)).$ In other words, a curve has an "orientation."

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I suppose it could be trying to suggest an equivalence on the set of all pairs, $(\phi,\psi)$ where, if there is homeomorphism $f:[s_1,s_2]\to [t_1,t_2]$ $f(s_1)=t_1, f(s_2)=t_2$ then $(\phi\circ f,\psi\circ f)$ is an "equivalent" curve. That's a very complicated, possible obtuse, reading of that sentence.

This would be saying that the integral along the curve only depends on the "order" in which the points of the curve are traversed, not the relative "speed" individual components are traversed.

I'd be interested in the full paragraph, and possible a few sentences back.