166 PROCEEDINGS OF THE AMERICAN ACADEMY 



The reduction of the observations for time and latitude is simple 

 enough, and the methods are given in the ordinary books. Some discre- 

 tion, however, is desirable in applying them. 



The application of least squares to time reductions is considered by 

 Struve often unnecessary ; nor is it generally practised in Germany 

 and Russia. Where it is applied, weights should be given to the ob- 

 servations depending upon the star's declinations. I am inclined, in 

 case the observations are fairly complete, and depend on about the 

 same number of wires, to consider the expression 



« Vl + secS^ 



as a fair representation of the probable error in different declinations. 

 Hence the weight will be expressed by 



2 



*"~l + secS2, 



that at the equator being taken as unity. If azimuth, collimation, and 

 clock-error, or rather small corrections of their adopted values, are the 

 unknown quantities, their co-efficients, multiplied by y' w, will be 



^ V^ = sin (cp-d) sec d J '^ =sin (cp-d) J ^ 



Y 1 -f" sec ■* V 1 -j- cos ^ 



C^ = J ? 



^ Vl+cos5 2 



^"~ Vl-f sec5« 



and their required squares and products 



^2 co = sin2(g) — 5) C^co 

 ^(7(0 = sin ((p—d)C^co 



(72(50= ^ 



1 + cos 52 

 Am =sin (cp — S) Con 

 2 



C 



CO 



00 



sec 5 + cos S 

 2 



■ 14- sec 52. 



I have tabulated the values of C-w, Ceo, and m, together with their 

 logarithms, according to these formula?, and give them in Table I. For 

 any station, the preparation of A^co, A Cm, Am, is at once very simple. 



The best results are not obtained from poor observations by cooking 



