198 STELLAE PnOTOGRAPHY. 



The data thus assumed for the position of the image are not the customary 

 rectangular co-ordinates, since the perpendicuUxr denoted by sin {t — v)smj) does 

 not intersect the axis of abscissas at the same point with the corresponding projection 

 of the instrumental parallel. Usually, however, these data will differ little from 

 rectangular co-ordinates, and they will serve in all cases to represent the general 

 course of the image of the star upon the photographic plate. The ordinary symbols 

 for rectangular co-ordinates will therefore be adopted, so that a; ^ cos (8 +^^), and 

 ^ = sin {t — v) sin 2)- 



From (1), cos p = sin (8 — y) + cos 8 sin 7(1 — cos {) ; this quantity progres- 

 sively increases from sin (S — y) when f = to sin (8 + y) when t ^= tt. The value 

 of z- must increase and diminish with that of cos p, unless S -^ j) is negative or 

 greater than tt. But 8 and p are always positive by supposition, and the greatest 

 value of p is | tt — 8 + y, which can never exceed v — 8 ; .r therefore increases 

 while t increases from to tt, returning through the same series of values as t 

 increases from tt to 2 tt, so that the value of x is the same for any two values of 

 i equidistant from or from v, no other value of x being identical with this. 



By definition, 



// = sin {t — v) sin p =^ sin t cos v sin p — cos t sin v sin p. 

 Hence, from (2) and (3), 



t/ = sin i sin y sin 8 + sin t cos t cos y cos 8 — sin t cos t cos 8 ; 

 and after reduction 



?/ = sin y sin 8 sin i — sin^ | y cos 8 sin 2 t. (4.) 



Differentiating this equation with respect to t, we have 



From (1), 

 we have also 



jj = sin y sin 8 cos t — 2 sin" -^ y cos 8 cos 2 t. 



d cos p . CN ■ / 



, = sin y cos o sm ? ; 



dt 



d X sin (8 -\-p) sin (8 -\- ])) 



d cos p sin p sin (8 -\-p — 8) cos 8 — sin 5 cot (8 -\- p)' 



Accordingly, 



d X sin 7 cos 8 sin t 



dt cos 8 — sin 8 cot (8 + ^>) ' 



and 



d y [cos 8 — sin 5 cot (5 -)- p)\ [sin y sin 8 cos t — 2 sin^ \ y cos 8 cos 2 t] 

 d X sin y cos 5 sin < ' 



