95 
investigation, where there seemed an opportunity of prose- 
cuting it further with success; and since the incorporation 
of observations made with the ring-micromete® had in- 
creased the probable errors of the results, and since the 
Series with the filar micrometer extended through the whole 
period of visibility excepting the observations by daylight, 
he passed on to still another determination from the filar- 
micrometer observations alone combined with the sextant- 
observations of February 28. From these he constructed 
eighty-three new equations of condition, determined a 
Seventh series of elements, reducing the probable error 
of a single normal to less than 8”.5, and assigned for each 
element its probable error. The period corresponding to 
these final elements was somewhat more than five hundred 
years, and it became a problem of much interest to deter- 
mine to what extent the resultant period might be varied 
consistently with the probable limits of errors of observa- 
tion. This Hupsarp solved most thoroughly by an ingeni- 
ous method of determining the variations of each of the 
elements, of the probable errors, and of each normal place, 
43 a function of the variation of the eccentricity, So that 
by substituting in these expressions the change of eccentri- 
city corresponding to any suspected period, a few minutes of 
figuring will give us the corresponding elements, the prob- 
able error of normal places, and the individual discordances 
of observations. This substitution he carried out himself 
for the period of one hundred and seventy-five years, and 
found that it implied a probable error of 113 for a single 
observation, and no individual discordances beyond the limit 
of reasonable error; although, to be sure, a certain “rate” 
seems indicated on this assumption by the earlier observa- 
tions, The limits of periodic time consistent with its ob- 
Served geocentric path were thus shown to be extremely 
