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Let S' be an x'y'-coordinate system. Let the x'-axis of S' coincide with the x-axis of an xy-coordinate system S, and let the y'-axis of S' be parallel to the y-axis of S. Let S' move along the x-axis of S with velocity v in the direction of increasing x, and let the origin of S' coincide with the origin of S at t = t' = 0s.

Let a ray of light depart from x'0 = 0m at the time t'0 = 0s towards x' = x'1 and reach x' = x'1 at the time t' = t'1, and let it be reflected back at x' = x'1 and reach x' = x'0 at the time t' = t'2. Let x'1 - x'0 = |x'0 - x'1| = L. Then with respect to the moving system S',

t'1 - t'0 = L/c, and

t'2 - t'1 = L/c = t'1 - t'0, and with respect to the stationary system S,

t1 - t0 = L + v*(t'1 - t'0)/(c + v) = t'1 - t'0, and

t2 - t1 = L - v*(t'2 - t'1)/(c - v) = t'2 - t'1.

Now let the ray of light depart from y'0 = 0m at the time t'0 = 0s towards y' = y'1 and reach y' = y'1 at the time t' = t'1, and let it be reflected back at y' = y'1 and reach y' = y'0 at the time t' = t'2. Let y'1 - y'0 = |y'0 - y'1| = L. Then with respect to the moving system S',

t'1 - t'0 = L/c, and

t'2 - t'1 = L/c = t'1 - t'0, and with respect to the stationary system S,

t1 - t0 = sqrt(sq(L) + sq(v*(t'1 - t'0)))/sqrt(sq(c) + sq(v)) = t'1 - t'0, and

t2 - t1 = sqrt(sq(L) + sq(v*(t'2 - t'1)))/sqrt(sq(c) + sq(v)) = t'2 - t'1.

The result of the Michelson-Morley experiment confirmed that

t2 - t0 = (L + v*(t'1 - t'0))/(c + v) + (L - v*(t'2 - t'1))/(c - v) = 2*(c*L - sq(v)*(t'2 - t'0/2))/(sq(c) - sq(v)) = sqrt(sq(L) + sq(v*(t'1 - t'0)))/sqrt(sq(c) + sq(v)) + sqrt(sq(L) + sq(v*(t'2 - t'1)))/sqrt(sq(c) + sq(v)) = 2*sqrt(sq(L) + sq(v*(t'2 - t'0/2)))/sqrt(sq(c) + sq(v)).

The result of the experiment is described as negative because it was expected to confirmed that

T1 = 2*L*c/(sq(c) - sq(v)) was not equal T2 = 2*sqrt(sq(L) + sq(v*(T2/2)))/c.

Moreover, the result of the Michelson-Morley experiment is actually independent of the value of v. Thus it did not reflect at all that the velocity v of the moving system S' with respect to the stationary system S is 0m/s, where S' is to be taken as the Earth and S as the assumed ether.

Let a ray of light depart from x'0 = 0m at the time t'0 = 0s towards x' = x'1 and reach x' = x'1 at the time t' = t'1, and let it be reflected back at x' = x'1 and reach x' = x'0 at the time t' = t'2. Let x'1 - x'0 = |x'0 - x'1| = L. Then with respect to the moving system S',

t'1 - t'0 = L/c, and

t'2 - t'1 = L/c = t'1 - t'0, and with respect to the stationary system S,

t1 - t0 = L + v*(t'1 - t'0)/(c + v) = t'1 - t'0, and

t2 - t1 = L - v*(t'2 - t'1)/(c - v) = t'2 - t'1.

Now let the ray of light depart from y'0 = 0m at the time t'0 = 0s towards y' = y'1 and reach y' = y'1 at the time t' = t'1, and let it be reflected back at y' = y'1 and reach y' = y'0 at the time t' = t'2. Let y'1 - y'0 = |y'0 - y'1| = L. Then with respect to the moving system S',

t'1 - t'0 = L/c, and

t'2 - t'1 = L/c = t'1 - t'0, and with respect to the stationary system S,

t1 - t0 = sqrt(sq(L) + sq(v*(t'1 - t'0)))/sqrt(sq(c) + sq(v)) = t'1 - t'0, and

t2 - t1 = sqrt(sq(L) + sq(v*(t'2 - t'1)))/sqrt(sq(c) + sq(v)) = t'2 - t'1.

The result of the Michelson-Morley experiment confirmed that

t2 - t0 = (L + v*(t'1 - t'0))/(c + v) + (L - v*(t'2 - t'1))/(c - v) = 2*(c*L - sq(v)*(t'2 - t'0/2))/(sq(c) - sq(v)) = sqrt(sq(L) + sq(v*(t'1 - t'0)))/sqrt(sq(c) + sq(v)) + sqrt(sq(L) + sq(v*(t'2 - t'1)))/sqrt(sq(c) + sq(v)) = 2*sqrt(sq(L) + sq(v*(t'2 - t'0/2)))/sqrt(sq(c) + sq(v)).

The result of the experiment is described as negative because it was expected to confirmed that

T1 = 2*L*c/(sq(c) - sq(v)) was not equal T2 = 2*sqrt(sq(L) + sq(v*(T2/2)))/c.

Moreover, the result of the Michelson-Morley experiment is actually independent of the value of v. Thus it did not reflect at all that the velocity v of the moving system S' with respect to the stationary system S is 0m/s, where S' is to be taken as the Earth and S as the assumed ether.

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