168 PROCEEDINGS OF THE AMERICAN ACADEMY 



the Gaussian logarithms. The best four-place table I know (J. H. 

 Traugott Muller's, 2d edition, Halle, 1860) has them in excellent shape 

 for this purpose. The micrometrical and refraction corrections should 

 be placed in one column, and computed together by a small table of 

 the value of one division or its logarithm, as affected by refraction at 

 various altitudes. A very trifling correction from the usual table is 

 necessary in Rocky Mountain work, as the barometer may stand at 23 

 or 24 inches instead of 30. 



The form of reduction which I suggest will be found in Lieut. 

 Wheeler's report on the geographical positions of Cheyenne and 

 Colorado Springs. 



In my catalogue of 981 stars, the logarithms of a' b' c' cV are given 

 for 1875, and will serve for some years to come. The trifle of error 

 introduced by their use after the lapse of a few years can best be 

 corrected by selecting some few stars, and computing their reductions 

 to mean place, say for 1975, thus getting the correction for 100 years ; 

 or by differential formulae. These will be 



da' Secular variation 



H 100 



dh' da ,15 . 



-77 :^ — cos « -r- ^ — aa'— sm 1" 

 dt dt n 



-r- = |_ — tan CO sm — sm a cos oj -^ — cos a sm -— 

 = — [^\b ad' -\- a c' ivixi 8 -\- \b a' df] sin 1" 



dd' . . ^ da , ^ d5 r. - ,■ . c. 



-77:= — sm a sm -^ -f- cos a cos -r- = [lo ao' sm 



a'- 



-j- — cos 5] sin 1" 



In computing the probable error of the latitude determinations, I 

 should proceed as follows : — 



The stars should be classified, and the probable error of the cata- 

 logue declination of each class estimated, as suggested above. The 

 comparison of observations of the same pair will give the probable 

 error of observation. The mean reciprocal of the number of observa- 

 tions on each pair should be taken : its reciprocal will give the average 

 weight of a pair as depending on this circumstance only. 



The pairs should now be classified by computing the probable error 

 to be expected, owing to both causes : those pairs which are once or 

 twice observed, or whose stars are both doubtful (Class C), will give a 

 large a priori probable error. These probable errors should now be 

 compared with the actual ones, to ascertain if any error constant to 



