First study the apparatus as it is actually set up and understand its operation. Questions are welcome. Then measure the path difference traveled by the two beams. A variety of devices can be procured for this task, depending on the accuracy you wish to obtain. You might first determine the accuracy with which you can measure time differences on the oscilloscope and, from that, determine how accurately you must measure the path difference.
The time measurement is to be made on an oscilloscope, utilizing
the linear sweep speed of the horizontal trace. The oscilloscope
can be set to display the waveform from both detectors on its screen,
one above the other, like the figure at right. The time delay,
, by which the
lower wave is shifted is the delay caused
by the longer travel path of the corresponding
light beam. The measurement of 
is the crucial one in this experiment.
To do this you can estimate the peak (or the valley) positions
or, if the waves are
centered vertically about an
horizontal line on the scope face, the zero crossing points
of the waves.
To get 
from the oscilloscope, position the vertical cursors on the scope
screen at the appropriate points on the waveforms and read the time
difference directly from the scope display.
The speed of light is then found from the measured path
difference and the time difference 
.
The accepted value for the speed of light is
m/sec.
This is its speed in a vacuum. The index of refraction of air at one
atmosphere is 
.
Does your accuracy justify a correction
for the index of refraction of air?