PHOTOGRAPHIC RESEARCHES NEAR THE POLE OF THE HEAVENS. 53 



Let us now introduce two new unknown quantities d x and d v de 



fined by the 



d n x = cos A J d n % — sin A J d n n, 

 d n y = — sinAjtfJ — cosA n 0rf n 7] 



Then, since we have, very nearly: 



(11) 



**. = *.,.— V, 



we get by substitution in equations (10): 



u \ -*- P s d n A ' + sin % d n x + cos B, s d n y + C' n? , = 



^X W ~ COS£ ^ n tfH-sin£ 0J X^C,, = = • 



* ; S 



. . (12) 



In these equations the coefficients no longer vary with n, but only 

 with 5. In other words, they are constant for all the impressions of the 

 same star. 



Equations (12) bring out the interesting point that C n , and £' . will 



n*s ^n%s 



be constants for any given star, provided d n x, d n «/, d n A\ and d n to are 

 constants. But if we square and add equations (1 1), we get: 



so that if d n x and d n y are constants, we have: 



d n | 2 h- d n vf = constant 



Now if £ n f and X! n are constants for each star, we shall have the 

 images of the various stars appear in exact concentric circles. Therefore the 

 above equations prove that if all the stars give exact concentric circles on 



the 



follow that the telescone has remained 



unmoved during the night. Rigorously, we shall satisfy the condition of 

 concentric circles if the shifting of the telescope has caused the pole to 

 move on the plate in a little circle at the uniform rate of 15° per hour, 

 and in the direction of diurnal motion. Nor is it theoretically possible to 

 determine the radius of this little circle. Practically, however, very com- 

 plete precautions will always be taken to make the telescope stand still. 

 If then we find the star-images in concentric circles, the conclusion is irre- 

 sistable that the pole has really not moved on the plate. To assume that it 

 moved in a circle in a definite direction at a definite uniform rate, is to 

 substitute a complicated and extremely improbable explanation of the observ- 

 ations for a simple and probable one. Later on, however, we shall examine 

 a little more closely the point touched upon here. 



*na.-MaT. wp. 53. 13 



