PHOTOGRAPHIC RESEARCHES NEAR THE POLE OF THE HEAVENS. 81 



d A denotes the mean of the values of d t A 

 advisable to treat the equations involving v 



be 



to treat the equations involving pd t to in the same way. The sub- 

 traction of each equation from the mean of all will remove the unknown z.. 

 It is also to be noted that in the least square solution, the equations derived 

 from the right ascensions must be multiplied by put sin l", in order to make 

 all the equations of equal weight. 



The period during which it is necessary for the photographic observa- 

 tions to be continued, in order to secure a determination of good weight, is 

 of course very different for the different unknowns. Thus a determination of 

 the precession would necessitate observations extending over a period which, 

 though considerable, should not approximate to that required by existing 

 methods. A very satisfactory determination of the nutation would require 

 the observations to extend through a period of nineteen years. On the other 

 hand, the aberration ought to come out with very good weight from one 

 year's observations only, if we assume the precession and nutation constants 

 known. In such a study of the aberration, we would simply omit z x and z y 

 Or we might determine the aberration as a function of the nutation and 

 precession constants, or of the nutation constants alone, in order to see how 

 much our work would be damaged by uncertainty in these constants. 



It may be of interest to add a few remarks on the instrumental appli- 

 ances to be used in such a research as has been outlined. It is clear that 

 though we have made it an essential principle of the method to determine 

 each night the path of the pole caused by motion of the telescope, it will ne- 

 vertheless be best to use a fixed telescope mounted on a special polar pier. 

 This telescope should be supported in Y's so as to permit of rotation about 

 its optical axis. We could then attempt to avoid one of the sources of uncer- 

 tainty to which a research of the present kind is liable. This is the so- 

 called optical distortion of the object glass. It is fortunately possible to 

 eliminate it in a measure, provided the telescope be mounted in the way we 

 have said, so as to permit of rotation about the optical axis. It will merely 

 be necessary each night, before beginning the exposures, to set the instru- 

 ment in such a way that a plane passing through the optical axis and any 

 given star, will always cut out the same section of the object glass. In thai 

 case, the optical distortion, both in right ascension and in polar distance, 

 will always be approximately constant for any given star. But we use in 

 our investigation only the variations of the right ascension and polar distance 

 with the time. Consequently, we shall very nearly escape the optical distort- 

 ion, for if it be constant, it will not affect the variations in question. 





$H3,-MaT. CTJ>. 81* 4 1 



6 



