42 



H. JAC0BY, 



r 



So much being premised, let us adopt the following notation: 



Let x, y, be the coordinates on the plate of any star, 



, 7), be the coordinates on the plate of the point corresponding 



to the pole of the heavens, 

 a, it, be the star's right ascension and polar distance on the sky 

 to, be the scale value in seconds of arc, or the number of seconds 



■ 



corresponding to one linear unit at the centre of the plate. 



We shall now introduce for convenience the auxiliary quantities p, B, 

 9, B', B'\ and A, defined by the equations: 



p sin B = y — Y], 





p cos B = x — 

 p sin 2?' ' = — y), 



? cos B' = — £, 



B" = B' 



A = an auxiliary constant. 





This having been done, we have the following equations for computing 



a and tz: 





a' = B-*-±U»* sin l" — ~p 2 sin B" cos B" 'to 2 sin V' + A, 



*' =i > to - j pV cos 2 5" w 8 sin 2 1" - ~ jj (a; 2 -*- ?/ 2 ) to 8 sin 2 1" + -i-i? 8 co 8 sin 2 l"" 



The demonstration of these equations can be obtained without diffi- 

 culty from the considerations contained in a former article on photographic 

 reductions (Astr. Jour. Vol. X, p. 129). They possess a precision quite 

 sufficient for the reduction of plates whose centres are not more than one 

 degree from the pole, and which have a radius of about one degree from 

 the centre of the plate. In the special case, however, where the centre of 

 the plate is within a couple of minutes of the pole, the radius of the plate 

 may be extended to two or even three degrees without introducing errors 

 greater than 0?01 or 0'.'02. 



Nothing has been said as yet concerning the orientation of the coordi- 

 nate axes on the plate. It will be found convenient to orient the plate so 

 that the positive axis of y will pass very nearly through the position of the 

 pole on the plate, and the positive axis of x point toward those stars whose 

 right ascensions are greater than the right ascension of the centre of the 

 plate. When, however, the pole is very near the centre, the axes should be 

 oriented so that the positive «-axis points toward 0° right ascension, and 

 the positive «/-axis toward 90° right ascension. In either case, A will be 



$*8.-MaT. CTp. 42. 2 



