Mr. Troughton on dividing Instruments. 123 
last variety, being the remainder of the table, consists of thirty 
two lines, and was obtained with a distance of 4,° 13' 7", 5. 
The table of apparent errors, or half differences, just ex- 
plained, furnishes data for computing the table of real errors. 
The rule is this ; let a be the real error of the preceding dot, 
and b that of the following one, and c the apparent error, taken 
from the table of half differences, of the clot under investiga- 
tion ; then is ~~ + c: = its real error. But, as this simple 
expression may not be so generally understood by workmen 
as I wish, it may be necessary to say the same thing less con- 
cisely. If the real errors of the preceding and following dots 
are both -j-, or both — , take half their sum and prefix thereto 
the common sign ; but, if one of them is -{-, and the other — , 
take half their difference, prefixing the sign of the greater 
quantity : again, if the apparent error of the dot under inves- 
tigation has the same sign of the quantity found above, give 
to their sum the common sign, for the real error ; but if their 
signs are contrary, give to their difference the sign of the 
greater for the real error. I add a few examples. 
Example 1. 
For the first point of the second quadrant. 
Real error of the first point of the first quadrant - 0,0 
Real error of the first point of the third quadrant — 6,9 
Half sum or half difference - - — 3,4 
Apparent error of the dot under trial - -^12,2 
Real error - - 8,8 
