198 Wisconsin Academy of Sciences, Arts and Letters. 



the three north of the zenith and above pole —1' 25". 183; and the zenith 

 distances are nearly equal for these two groups. It is plain from this that 

 the least square value —1' 24". 888 is probably nearer correct than the 

 greater (algebraic) value —1' 24". 803; as it ought to be. In other words, 

 the exclusion from the final clock corrections of the quick moving polar 

 stars renders the elimination of pivot error very questionable; as, in fact, 

 does the employment of groups not sufficiently well-balanced. 



The simpler method of reduction, which gives the instrumental correc- 

 tions separately, without least squares, really assigns to the iDolar stars their 

 proper weight; especially if they are aU observed at upper culmination, as 

 the German instructions recommend. For in this case the final clock cor- 

 rections from each time star are actually reduced by interpolation to the 

 zenith, as may be seen by using Dr. Braun's graphical method given in 

 vol. 109 (No. 2595) of the Astronomische Nachrichten. So that in the pre- 

 vious example it is not surprising that the i>reliminary solution gave a 

 value (1' 24". 90) very nearly equal to the final result. Nor is it very plain 

 that the least square method is absolutely indispensable; I have employed 

 it in my reductions without objection, although I ha^e never known a case 

 in which it materially aided in producing a better agreement in the result- 

 ing longitude. The chief argument in its favor here is that it renders it 

 unnecessary to restrict the selection of stars to a narrow range, while it re- 

 moves the arbitrary character of the reductions when this restriction has 

 not been carried into effect. The chief objection to the method of least 

 squares is that observers who are not both experienced and careful some- 

 times permit a blind faith in it to mislead them, in preparing their working 

 Usts; and forget the necessity of making observations enough and of the 

 right quality, which is no less when least squares are used in reductions 

 than at any other time. 



Jacobi's theorem is in substance the following; and bears directly upon 

 the point in question: 



In order to obtain the least square result for any unknown of m equations 

 with a less number of vmknowns, we solve all possible combinations n by n 

 of the m equations, and multiply each such result for this unknown by the 

 square of the corresponding determinant. We add all these products to- 

 gether and divide their sum by the sum of the squares of the determinant 

 factors; the quotient wiU be the least square result sought for.^ 



To a least square result, then, unfavorable combinations furnish small 

 contributions relatively to favorable ones; and if the favorable ones cannot 

 be made, the unfavorable ones are better than nothing. Stars below pole 

 are the proper ones to employ in combination with those above, when the 

 azimuth is the unknown most needed, as in setting uj) a meridian mark; 

 but for time-determinations proper the best combination is that of polars 

 above pole with time-stars at nearly equal zenith distances on the other side. 



•> Jacobi, De Formatione et Proprietatibus Determinantium: Crelle's Journal. 



