Measurements of Rulings on Glass. By E.W. Morley. 141 
probable error of a mean. But the probable difference of the two 
measurements of one and the same interval will afford a convenient 
and, for the purpose, sufficient means of estimating the degree of 
confidence which may be felt in the result. The differences between 
two measurements of the same interval were one-fourth of a diyi- 
sion in seven cases, two-fourths in five cases, three-fourths in three 
cases, one division in two cases, one division and one-fourth in four 
cases, and one division and two-fourths in two cases. From this it 
appears that the probable difference of the two measurements of the 
same interval is fifty-six hundredths of a division of the screw head. 
Now the value of one revolution of the screw in the given circum- 
stances was the fourteen hundred and seventy-sixth of an inch, or 
0-0006775. Hence the probable difference of two measurements 
for the same interval was the two hundred and sixty-one thousandth 
of an inch. It will be seen that this degree of accuracy was amply 
sufficient for the purpose. It will be noticed that twelve of the 
actual differences were smaller than the probable difference, and 
eleven were larger. 
In Figure a, Plate CLXXV., the measurements are plotted by 
making the abscissze proportional to the distance of the initial line 
of each measurement from the line where the measurements began, 
while the ordinates are proportional to the differences between the 
successive measurements and a constant subtrahend. It will be 
seen that an error whose period is five times the measured interval 
is clearly indicated. 
If the shortest interval of each of these five cycles is subtracted 
from the longest, the differences are successively seventy, eighty- 
six, fifty-five, fifty-eight, and fifty-eight: the unit being the four 
hundredth part of the revolution of the micrometer screw. Half 
of the mean of these differences is that part of the periodic error 
which corresponds to the fifth part of the circumference of the screw 
of the ruling engine, or to an are of seventy-two degrees. This 
quantity is eight and seventeen hundredths divisions of the screw 
head of the micrometer. If this be multiplied by the ratio of the 
diameter of the circle to the chord of the arc or seventy-two degrees, 
we have a tolerable approximation to the whole periodic error of 
those threads of the screw which produced these lines, as the screw 
was at the time adjusted. This quantity is the ten thousand six 
hundredth of an inch, or 0000094 inch. This quantity represents 
the greatest possible displacement of a line from its true place, as 
far as the displacement is periodic and not accidental; the greatest 
possible difference between two professedly equal intervals is double 
this quantity. ' 
If doubt is felt as to the propriety of assuming that the errors 
for a fifth ofa revolution and for a half revolution are proportional 
to the chords of the ares, the total displacement of a line by periodic 
