400 
HOWARD E. PULLING 
When 7 = i8o°, sin 7 = 0, sin a = 0,0; = o and there is no error. 
It is apparent that the error is a maximum when 7 = 90°, and in that 
case sin a = n/R, and since n = o.i mm. and R = 17.5 cm., 0:2 = a 
Httle less than 2^ That is, the difference between the observed angle 
and the true angle will be a little less than 2' of arc. 
Since a 15° protractor has been found to have sufficient range for 
all ordinary purposes, the illustration may be made more applicable 
to actual conditions if an angle of ten degrees is considered. Suppose 
this angle to be measured between 80° and 90° and thus at the point 
of maximum error. For both 80° and 90° jS > 7, and hence {^2 — 
— (72 — 7O = the angular error. /32 = 72 + ^2 = 90° 2'; ai = (n/R) 
•sin 80° = 1° 57'; /3i = 71 + «i = 80° i' 57'^ 72 - 71 = 10°; - /3i 
= 10° 3'^ The angular error is, then, 3". 
The linear error, y, is found from the equation 
. ^2—^1 . 72 — 7i 
2r-sm — 2r-sm = 7. 
2 2 
With the combination of lenses (No. 2 ocular and 1/12 oil-immersion 
objective) with which the micrometer has been chiefly used, r = 
0.030885 mm. When this value is substituted in the equation y is 
found to be 0.000001 mm., a negligible quantity. Furthermore, y 
will decrease with increasing magnification because f , which decreases 
with increas'ng magnification, enters the equation as a factor. 
Case 2: Line joining the two centers on the zero radius of the 
vernier. In principle this is case i. An angle of 10° measured 
between 0° and 10° would be the same as an angle of 10°, between 80° 
and 90°, o" case i. The same conclusions hold for both. 
II Failure to Have the Radii of the Protractor and Vernier 
THE Same 
Only the case in which the vernier radius is longer than that of 
the protractor need be considered. The reverse would cause the pro- 
tractor either to bind in turning or to overlap the vernier, and either 
difficulty would at once be apparent. If the end lines of graduation 
of vernier and protractor lie on the same straight line and both 
members are accurately graduated, the angle read will be the true 
angle regardless of the length of the protractor radius. If they do 
not lie on the same straight line the error may be detected by inspec- 
tion. 
