56 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
minimum of computation will be required if the observations can 
be so arranged as to eliminate the effect of these factors in the final 
value of the correction to the time-piece. Although it is usually im- 
possible to eliminate the effect of the azimuth, collimation, and rate 
completely, it is generally possible to make a close approximation 
thereto. To show this fact analytically let 
= the correction to the time-piece at the epoch ¢,; 
= the observed time of a star’s transit; 
= the star’s right ascension; 
= the azimuth of the transit; 
= the collimation of the transit; 
= the rate of the time-piece; 
the azimuth factor ; 
= the collimation factor; 
= the weight of (¢ — 4); 
= the residual error. 
Saif. 3 fen 
| 
Then the type observation-equation will be 
at+ Aat+tCce+t—t)r+ti-—-a=v». (1) 
The normal equation in At, using brackets to indicate summation 
of like quantities, is 
[p]4t+[pA]a+[pCl]e+[pG@—t)]r+ [pG—a1= 0. (2) 
~ This shows that in order to secure the complete elimination of 
the effect of a, c, and r, we must have 
[pA] = 0, [pC] = 0, [p ¢@ — t.)] = 0. (3) 
The last of these conditions can always be fulfilled by making 
fe Lee) 
bij seit re 
It may be shown that the value of A¢ corresponding to ¢, as 
defined by (4) has a maximum weight. A close approximation to 
the first two conditions of (3) can be secured by selecting for obser- 
vation stars of suitable declinations and by reversals of the tele- 
scope. 
If we put 
Load sy ee Lee 
behets) aL se? 
equation (2) gives At = — At, — Ba—jye. (5) 
This shows that in case # and y are small, as supposed above, an 
