SURVEY OF MASSACHUSETTS. 49 
“The formule used are found in Malortie’s Topography, vol. i, p. 181—135. They 
are as follows. (I have changed a few of the symbols for greater convenience.) 
“1st. To reduce the angles as observed to the tops of the signals. At the point A, 
(Fig. 5,) an observer with the telescope at a observes the zenith dis- 
tance* (= 90° + the angle of depression or elevation, as the case may 
be,) of the top of the signal b’ at the point B. At B, with the telescope 
b, he observes the Z. D. of the top of the signal a’ at the point A. It 
is obvious that the angles, measured at the points a and b, are eccen- 
tric, and must be reduced to what they would be at a’ and b’, before 
they can be used in computing the difference of level between the 
points A and B.t ‘To effect this, let the distance aa’, or the differ- 
ence in height between the top of the signal and the telescope at the 
station A, be represented by ; the Z. D. measured at a by A, and the 
arc AB in ft, by K, then 
(1.) i (2) d=Aad 
And the same at the station B, where » = bb’; A the Z. D. measured at b, and & = A+d. 
d, being the correction in seconds to be applied to the observed angle. 
“2d. To find the mean refraction in seconds. Let 7 = refraction; CO the angle at the 
centre of the earth, or, which is the same thing, the arc A B contained between the two 
stations, expressed in terms of the arc, and let 6 and 8’ = corrected zenith distances; then 
(3) re c— (cea) 
“This formula is founded upon the proposition, that if the sum of the two depressions 
be taken from the measure of the intercepted terrestrial arc, half the remainder is the 
refraction. If one of the objects is elevated, on account of the smallness of the arc, then 
if the depression be taken from the sum of the contained arc and elevation, half the 
remainder will be the refraction. The value in seconds of the are A B may be found, 
with sufficient accuracy, by using the mean radius of the earth as given in the books. 
“3d, To find the difference of level. Let H represent the difference of level between 
the tops of the signals at A and B; then 
_ Ksin 5 (0 — 9) 
~ cos 2 (5 —d +0) 
(4.) H= K tan =: or; (hi) 
“The first of these two formule, although not exact, is yet sufficiently accurate for all 
practical purposes, and is the one which has been used in our calculations. 
* We shall, hereafter, instead of zenith distance, frequently use the initials Z. D. 
+ The angles may, in like manner, be reduced to the height of the telescope of the instrument at the respective sta. 
tions, or to the copper bolts which mark the stations, whichever may best suit the notions of the computer, 
VOL. IX.—16 
