STELLAR rnOTOGRAPnV. 197 



path of the star with respect to the telescope. The easiest supposition respecting 

 the inclination of the plate to the polar axis of the telescope will be that this 

 inclination is equal to 8 ; that is, 'we may assume that the telescope is set to the 

 declination of the star, without regard to error of adjustment. If the clock driving 

 the telescope is correctly regulated, which is here assumed to be the case, the plane 

 of the plate is constantly perpendicular to the plane of an instrumental hour circle, 

 which makes the angle t with the fixed circle. The orthographic projection of this 

 hour circle upon the plate is a straight line, which may be regarded as an axis of 

 abscissas. The intersection of this line with the radius of the sphere perpendicular 

 to it may be assumed as the origin ; the point of the sphere to which the radius 

 is directed is at the distance \ tt — 8 from the instrumental pole, which will there- 

 fore be orthographically projected upon the plate at the distance cos 8 from the 

 origin. The orthographic projection here employed has the advantage of simplicity; 

 it will not, however, correctly represent the relative dimensions of different parts of 

 the curve described upon the plate by the image of the star, iniless y is very small. 

 But the present inquiry is confined to the general form of the curve, which may be 

 derived as well from the orthographic projection as from one more strictly appro- 

 priate to the circumstances of the case. The gnomonic projection would probably 

 best repi'esent the actual cui've. 



The small circles formed by the intersections of the sphere with planes perpen- 

 dicular to the axis of the instrument may be called instrumental parallels. Their 

 orthographic projections upon the plate will be elliptical, but the parts of these 

 projections near the origin will in ordinary cases differ little from straight lines. 

 The projection of the instrumental parallel of the star will intersect the axis of 

 abscissas at two points, the distances of which from the origin may be expressed by 

 sin (1^ TT — 8 — p) and sin (| tt — S + />}. The first of these quantities, which is equal 

 to cos (8 -\r p), will here be employed as one of the data indicating the form of the 

 curve which is described upon the plate by the image of the star. The employment 

 of the second quantity is never necessary, although, when \ tt — 8 < y, it may some- 

 times seem more appropriate. 



The least distance of the imaoje of the star from the axis of abscissas will be 

 expressed by sin {t — v) sin p. This will appear upon consideration of the case in 

 which 8=0, and the inclination of the plate to the axis of the instrument will not 

 affect the length of the pi-ojection, provided, as has been supposed, that the plate 

 remains perpendicular to the instrumental hour circle at the angle t from the fixed 

 circle. 



