The equations used to describe the transmission of light up to the time when Albert Einstein published his Special Theory of Relativity in 1905 were the Galileian transformation equations:
x'=x-vt
y'=y
z'=z
t'=t
These equations were considered proven wrong after the Michelson-Morley experiment because that experiment showed that the speed of light measured the same regardless of motion of a source of light or an observer. Einstein described this relationship by two little equations which he extracted from equations used by H.A. Lorentz to describe electromagnetism.

x=ct
x'=ct'
Comparing these two little equations with the Galileian transformation equations, we see that the second of these equations cannot be used in this form with the Galileian transformation equations because the term t' was already used in the equation t'=t. In order to express that light travels a distance of x' as shown in the Galileian transformation equations, we have to use some other term than t'. We will therefore use t2, meaning that we will measure this time according to a clock in K, the system at rest.

x'=x-vt
x'=c(t2)
c(t2)=ct-vt
t2= t(c-v)/c
We can then describe the transmission of light by the following equations.

c=x/t = x'/[t(c-v)/c] = (x-vt)/[(ct-vt)/c]

= (x-vt)/(t-vt/c) = (x-vt)/(t-vx/c^2)

= (x-vt)gamma/(t-vx/c^2)gamma

gamma= 1/sqrt(1-v^2/c^2)

(x-vt)gamma/(t-vx/c^2)gamma = x'Lorentz/t'Lorentz

We see from this that as it pertains to times and distances in the Galileian transformation equations, Einstein's second equation should read

x'Lorentz=c(t'Lorentz)

since x' in the Galileian transformation equations is a shorter distance than x'Lorentz, meaning there would have to be a distance contraction to correct if x'Lorentz were to be substituted directly for x' as a distance.
We therefore consider the distance x' as it applies to a moving system K'. According to the above equations, if we multiply x'Lorentz by
sqrt(1-v^2/c^2) we will get the distance x'. This is presently done by scientists to adjust for the distance contraction caused by substitution of x'Lorentz directly for x' and is called the Lorentz-Fitzgerald contraction.
Robert B. Winn