320 
Proceedings of the Royal Irish Academy. 
It will be observed, that Savary's elements represent tlie observa- 
tions up to '^0. 6 very well. After JN'o. 6, the calculated position 
falls behind the observed position, the difference reaching a maximum 
in IS'o. 13, where it amounts to 9°. 44 ; the discrepancy then decreases 
until in ISTo. 22 the difference only amounts to 3°-62. Savary's ele- 
ments, though the earliest in point of date, do not exhibit such large 
discrepancies as either of the three remaining orbits. IS'o observations 
were available for the determination of Savary's orbit later than 
m. 7. 
The elements given by Herschel II. give for ISTo. 22 a calculated 
position 17°. 43 in excess of the observed position ; with this excep- 
tion, the greatest difference is 6°-06 in l^o. 13. With reference to 
this determination, Herschel II. remarks, *Hhe case is one highly 
unfavourable for the application of my method ; and, moreover, the 
resulting elements are those of a first approximation only." In the 
computation of these elements, as in those of Savary, no observations 
later than l^o. 7 were available. 
Madler's elements represent the first eighteen observations with 
surprising fidelity, and there is no very serious discrepancy until the 
last observation, I^o. 22, when the calculated position exceeds the 
observed position by 33°. 
Villarceau's elements are much of the same character as Madler's, 
and the differences (though greater than Madler's) are not very serious 
until JN'o. 22, where the calculated position is 37°* 91 greater than the 
observed position. 
If, as there cannot be a reason for doubting, one star of the pair 
has a relative motion about the other, in an ellipse of which the 
fixed star is the focus, it can hardly be maintained that any one of 
the four sets of elements which have been examined represent that 
ellipse with sufiicient accuracy. 
In the absence of Mr. Briinnow's observation of 1872*28 (I^o. 22), 
Madler's elements would, no doubt, have fair claims; but that obser- 
vation (in addition to those which immediately precede it) necessitates 
some change in the elements. 
The extremely elegant method invented by Sir John Herschel, 
and described by him in his memoir already referred to, appears a 
most appropriate method for deducing, at all events, approximate 
elements. It is proposed by the author of the present paper, to ap- 
ply this method to a new determination, embracing, for the purpose, 
observations between 1781*97 and 1872-28. The application of the 
method to the present instance will be described in detail. 
A sheet of paper neatly divided into square millimetres, by ruled 
lines, with every tenth line darker than the rest, is mounted upon a 
drawing-board. 
On a horizontal line abscissae are marked at the rate of two milli- 
meters per annum, corresponding to every date in Dawes' list of positions 
of I Ursae, already referred to. To these are added a few positions 
quoted by Herschel, not included in Dawes' list, and from Ko. 1 8 to 
