198 



STELLAR PHOTOGRAPHY. 



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

 rectangular co-ordinates, since the perpendicular denoted by sin (t — v) sin p 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 x = cos (8 -f-j»), and 

 y = sin {t — v) sin p. 



From (1), cos p = sin (8 — y) -f- cos 8 sin y (1 — cos t) ; this quantity progres- 

 sively increases from sin (8 — y) when t = to sin (8 -|- y) when t = w. The value 

 of x must increase and diminish with that of cos p, unless S -{- p is negative or 

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



value of p is £ it — 8 + y> which can never exceed ir — 8 ; x therefore increases 

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



increases from tt to 2 7r, so that the value of x is the same for any two values of 

 t equidistant from or from it, no other value of x being identical with this. 

 By definition, 



y = sin (t — v) sin p = sin t cos v sm p — cos t sin v sin p. 



Hence, from (2) and (3), 



y = sin t sin y sin 8 -f sin t cos t cos y cos 8 — sin t cos t cos 8 ; 



and after reduction 



y = sin y sin 8 sin t — sin 2 \ y cos 8 sin 2 t (4.) 



Differentiating this equation with respect to t, we have 



From (1), 



jj = sin y sin 8 cos t — 2 sin 2 \ y cos 8 cos 2 t 



d cos p cv • . 



^ t = sin y cos dsm/; 



we have also 



<? x _ sin (5 4- j») sin (5 -(- p) 1 



d cos jt? sin jt> sin (5 -|- jt> — 3) cos 5 — sin 3 cot (5 -f- P) ' 



Accordingly, 



d x sin y cos 5 sin t 



and 



c? « cos 3 — sin 8 cot (3 -f ^>) ' 



dy _ [cos 3 — sin 8 cot (3 -f- p)~\ [sin 7 sin 5 cos t — 2 sin 2 £ 7 cos 5 cos 2 *] , 



c? <c 



sin 7 cos 3 sin t 



y 



