MATHEMATICAL SECTION. 57 
approximate value of At is — At,. After some preliminary obser- 
vations at a station it is easy to render a and ¢ small, and their ap- 
proximate values may always be found from the observation equa- 
tions by a brief inspection; so that with such values of a, c, #, and 
y as are nearly always readily attainable At may be derived from 
(5) to the nearest 0°.01. 
We may thus dispense entirely with the other three normal equa- 
tions and reach the same result which would follow from their use. 
The solution may also be checked; for by one or two approxima- 
tions the values of At, a and ¢ which make [pv] = 0 can be readily 
found. 
The practical steps in deriving 4¢ from (5) may be summarized 
as follows: 
1. The mean of the observation-equations for clamp west minus 
the mean of those for clamp east will give an approximate value of 
the collimation c. 
2. The application of this value of ¢ to each observation equation 
will give a corrected value of (¢ — a) for each star. 
3. An approximation to the value or values of the azimuth will 
then result by eliminating At from one or more pairs of the corrected 
observation equations. The azimuth may then be applied to correct 
the values of (¢ — a), reached in step 2. 
4. The approximate values of a and ¢ will now give an approxi- 
mate value of At from (5), and the application of this value of At 
to the values of (t — «), derived in step 3, will give approximate 
values of v. 
5. Form [pv]. If this sum is not zero within 0°*.01 or 0°.02, a 
brief inspection will show what changes in a and ¢ (and _ possibly 
At) will make it zero within those limits. 
By this process of determining the residuals or their approximate 
values as soon as possible in the computation any large errors in 
the values of (£ — a) or the azimuth and collimation factors will be 
easily detected. 
In precise longitude determinations it is customary to have for 
each night’s observations two complete time determinations, one 
immediately preceding and one immediately following the tele- 
graphic comparison of time-pieces. In this case there will be two 
values of A¢. Calling these At’ and At’ and denoting the corres- 
