PHOTOGRAPHIC RESEARCHES NEAR THE POLE OP THE HEAVENS. 43 



the approximate right ascension of the point toward which the a?- axis is 

 directed. 



The proceding equations (1) will enable us to compute the right ascen- 

 sion and polar distance of any stars whose coordinates x and y have been 

 measured, provided we know the four constants: 



, Y], (D, A. 



These constants cannot in general be known accurately in advance, 

 but we shall always be in possession of very good approximate values of 

 them. If with these approximate values of the constants we compute the 

 right ascensions and polar distances of the stars, we can obtain readily the 

 necessary corrections by differential formulas. Thus if the approximate 

 values of §, y], w, and A require the corrections d£, dv\, da, dA, the corre- 

 sponding corrections da. and dit required by a! and it' will be found 

 differentiating equations (1). In doing this we need only take account of 

 the quantities B } A, and pu, because the other terms are so small that 

 they can always be computed with complete precision by means of the 

 approximate values of the constants. We thus find, remembering that: 



7 r» sio B jr cos B j 



dB = — r— -rf. d\ i—Tw dri, 



p sin \" n p 31U 1" " 



dp = — cos Bd% — sin B dt\, 



the following corrections: 



dn =pd(n — o> cos Bd\ — to sin Bdri 



Denoting now by a and it the corrected values of the right ascension 



and polar distance, so that: 



a = a' -*- dec, 



it = it 



— §— uit, 



we have: 



7t = it' -+-pdia — (o cos Bd% — to sin Bdt\ 



(2). 





Equations (2), which are our fundamental equations, can be used either 

 for calculating the a and it of unknown stars, or for determining the values 

 of c$, rfv), do, and dA from the measured coordinates of known stars. The 



*Ha.-Ma?. cTp. 43. 3 



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