A particle appears to be at rest when observed from a frame S_1 ,...

Let the velocity of the particle with respect to frame $S_1$ be $\vec{v}_1$ and with respect to frame $S_2$ be $\vec{v}_2$.

Given that the particle is at rest in $S_1$, we have $\vec{v}_1 = 0$. Thus, its acceleration in $S_1$ is $\vec{a}_1 = 0$.

The particle travels with a constant velocity in $S_2$, so $\vec{v}_2 = \text{constant}$. Thus, its acceleration in $S_2$ is $\vec{a}_2 = 0$.

The acceleration of the particle with respect to $S_1$ is $\vec{a}_1 = \vec{a}_p - \vec{a}_{S1} = 0$, which implies $\vec{a}_p = \vec{a}_{S1}$.

The acceleration of the particle with respect to $S_2$ is $\vec{a}_2 = \vec{a}_p - \vec{a}_{S2} = 0$, which implies $\vec{a}_p = \vec{a}_{S2}$.

Therefore, $\vec{a}_{S1} = \vec{a}_{S2}$.

If $S_1$ is an inertial frame, then $\vec{a}_{S1} = 0$. This implies $\vec{a}_{S2} = 0$, making $S_2$ an inertial frame as well.

If $S_1$ is a non-inertial frame, then $\vec{a}_{S1} \neq 0$. This implies $\vec{a}_{S2} \neq 0$, making $S_2$ a non-inertial frame as well.

Hence, both frames must be either inertial or non-inertial.