PHOTOGRAPHIC RESEARCHES NEAR THE POLE OF THE HEAVENS. 59 



In using equations (14) and (17), it is important to remember that 

 they hold true only on the condition that a and -it denote the apparent 

 right ascension and polar distance, as we see them on the sky. Now as the 

 expressions for u g aud v s involve a and it, and as the refraction and aber- 

 ration vary during the night, it is clear that u and v will not mean the 



different 



* s 



To avoid this difficulty, and 



the equations hold when a and iz are the true instead of the apparent co 

 dinates, it is merely necessary in the calculation of £"„ . and f _ . to add 



s 'n,s 



them the refraction and diurnal aberration corrections corresponding to the 

 sidereal times 6 B , and affected with the proper sign for turning the appa- 



■ 



rent a and it into the true a and iu. This having been done in computing 

 the terms £" n f and C n s of the observation equations, the values of the un- 

 knowns will be ready for further discussion. 



The annual aberration may be disregarded except in so far as the 

 slight change in the sun's longitude during the night may affect the annual 

 aberrations of the several exposures differently. We can easily compute the 

 amount of this little correction as follows. 



Put: 



do = Increase of Sun's longitude from the time o to the time 6 n . 



Then we shall reduce all our observations to the time o , by applying 

 the following corrections: 



For it — tc, -*-D cos z sin a dQ sin 1 



n 



G sec e cos a dQ sin l", 



For (otj — a) it sin l", -+- B cos £ cos a dQ sin 1 



n 



Csec s sin a dQ sin 1, 



where G, D, and £ have their usual meaning. 



The theoretical probable errors of the various unknown quantities may 

 be determined as follows. From the residuals obtained in the solution of 

 equations (17) we obtain the probable errors of the quantities A n # and k n y. 

 The corresponding probable errors of d' n l and d' n -t\ then result from a con- 

 sideration of equations (19). Knowing the probable errors of d' n l and d' n % 

 those of d^l and d 'r\ follow at once. We thus have the materials for com- 

 puting the probable errors of X" n and X' n from a consideration of equations 

 (25). These will then be the probable errors of equations (26), and the 

 weights with which these equations determine the y's being known, we at 

 once arrive at a knowledge of the theoretical probable errors of the y's. 



o 



Another but less accurate determination of the probable errors of the y 

 results from the residuals obtained in the solution of equations (26). Know 

 iag the probable errors of the y's, equations (24) will enable us to comput 



*n3.-Ma T . cm 59. '9 



