134 ROGERS AND McLEOD ON THE 



Every star obseryed will give an equatiou of the forms (2) or (B), according to the 

 method employed in the determination of the collimation. The conditions of this special 

 problem are such, however, that the best determination of the unknown quantities 

 involved will be had, when the stars observed are distributed in groups arranged in the 

 following order with respect to the declination. 



C (1) Stars having a large south declination. 



Time stars. < (2) Stars situated near the equator. 



( (3) Stars symmetrically distributed about the zenith of the observer. 



Tj , , ( (1) Stars at the upper culmination. 



JrOIiAR STARS. -, ■ ii i i • ■ • 



( (2) Stars at the lower culmination. 



Employing numerals to designate the time stars, primes to designate polar stars at the 

 upper cvilmination, and seconds the polar stars at the lower culmination and writing 



j„=[(r+« + 7?i)-«] 



we shall have a series of equations of the form : 



C4) 



1. = j;+yi,a + c;c + jr. 



2. = J„^-(r A^a + C„c + AT. 



3. = /!,''+ A]a +C,c -\- J T. 



4. = /l„' + A'a +C'c -{- /IT. 



5. = J; + A"a + C"c + JT. 



The fact that the values of the unknown quantities derived from equations of form 

 (4) are to a certain extent indeterminate must be recognized at the outset. 



The degree of precision with which they can be determined will depend largely 

 upon the proper distribution of the stars observed. If one does not inquire into the 

 nature of the problem, there would seem to be no reason why a solution of a series of 

 equations of the form given, by the process of least squares should not give the most 

 probable value for each of the unknown cjuantities, but it is to be noted that the limited 

 indeterminate character of the equations is inherent in the problem, and it can hardly be 

 assumed that this indeterminateness will be removed by a least square solution. 



If, however, the values of the constants a and c remain invariable throughout the 

 entire series of observations, this method of discussion can be most advantageously em- 

 ployed, especially since by the change of sign in the co-efhcient C, made necessary by the 

 reversal of the instrument, the divisor, through which c is determined, becomes large. If, 

 however, either a or c undergo a change in A'alue at an iniknoicn point of time during 

 the observations, the values of the' unknown quantities derived from a least square solu- 

 tion will be to a certain extent illusory, since the effect of the solution will be to distribute 

 an error, which really occurs at one point, over the entire series, in such a way as to do 

 the least harm. 



If th.e change in the value of either a ore takes place during the operation of reversal, 

 a solution of the entire series of equations under the supposition that a and c remain 

 constant, will generally give residuals for the polar stars which have opposite signs for 

 the two positions of the instrument. 



If it is found to be necessary to determine the constants for each position of the in- 

 strument separately, the accuracy of the determination will, to a certain extent, depend 

 upon the extent to which the conditions stated above are fulfilled, but some doubt must 



