STELLAR PHOTOGRAPHY. 195 



We must next consider the trail described by stars in other portions of the 

 heavens, supposing the only error to be that of the axis of the instrument. If it 

 were possible to photograph a star at the south pole without disturbing the instru- 

 ment, evidently it would also describe a circle like that described by a star at the 

 north pole, since, if the photographs were taken simultaneously, both stars would 

 alwaj'S be exactly 180" apart. The path of other stars may be determined by 

 conceiving of two concentric spheres revolving with equal velocity, the outer one 

 around an axis parallel to that of the earth, the inner one parallel to the axis of 

 the instrument. Tlie motion of any point of the outer sphere compared with the 

 inner one will give the required trails. Since the inclination of the axis is assumed 

 to be small, the effect will be the same as if the inner sphere remained at rest and 

 the outer one was moved so that its poles should describe circles around the axis 

 of the inner sphere, but without rotating either sphere. This could be accom- 

 plished mechanicall}'^ by loading the lower half of the inner sphere. If we consider 

 a point upon the equator of the outer sphere, we see that its motion will be exactly 

 north and south, any tendency of the northern pole to move it east or west being 

 always compensated by an equal tendency of the southern pole to move it in the 

 opposite direction. For other declinations, the east and west motion will be reduced 

 by an amount that will always be proportional to the sine of the declination, while 

 the north and south motion will be undiminished, and will always be equal to twice 

 the deviation of the two axes. In the case of stars, it therefore follows that the 

 trails will always be ellipses having their transverse axes north and south. The 

 length of the semi-transverse axis will equal the distance between the pole of the 

 instrument and the north pole, when represented on a sphere having a radius equal 

 to the focal distance. The semi-conjugate axis at the pole will be the same as the 

 semi-transverse axis; it will become zero at the equator, and will vary between these 

 points as the sine of the declination. The ti'ails will therefore vary from circles at 

 the pole to lines running north and south on the equator. If errors are present, 

 due to the rate of the clock as well as to the position of the polar axis, each will pro- 

 duce its effect independently. The form of trail may therefore be constructed 

 geometrically from the principles described above. For testing the instrument, the 

 pole is the best region, since errors in altitude and azimuth in the axis here enter 

 with their full force. At the equator an error in altitude is insensible, as the star 

 is there describing the end of a very elongated ellipse. 



To adjust the axis, it may sometimes be found more convenient to replace the 

 sensitive plate by a positive eye-piece with cross-lines. Directing the telescope 



