Astronomy. — “Approximate formulae of a high degree of ac- 
curacy for the relations of the triangles in the determination 
of an elliptic orbit from three observations.” By J. Werper. 
(Communicated by Prof. H. G. vAN DE SANDE BAKHUYZEN.) 
The places in space occupied by the observed planet or comet at 
the instants ¢,, ¢, and ¢, are indicated by P,, P, and P,, the posi- 
tion of the sun is indicated by Z. 
For the determination of an elliptic orbit we mainly proceed as 
follows: first by means of successive approximation we derive the 
distances P,Z=r,, P,Z=r,, P,Z=r, from the data of theobser- 
vations, from which distances the elements of the orbit are directly 
computed without using the intervals of time. From the obtained 
ellipse we can again derive the intervals of time in order to test the 
accuracy of the results and compare them with the real ones. In 
case they perfectly agree, the ellipse found satisfies all the conditions 
of the problem, but as a rule this is not so. The cause of it is 
that, in order to calculate the distances 7,, 7,, and 7,, we use 
: : ; triangle P ZP, 
approximate formulae to express the relations — — 
triangle P,ZP, 
triangle P,ZP, 
triangle P.ZP, 
three distances to be found, while neglecting the terms of the 294, 3*4 
or 4» order with respect to the intervals. Indeed, different expressions 
have been proposed for n, and #,, some recommending themselves by 
greater simplicity, others by greater accuracy, but, so far as I know, 
in the general case of unequal intervals none of them contain the 
quantities of the fourth order with respect to the intervals. 
De 
and == n, in terms of the intervals of time and of the 
The errors in the calculated distances 7,,7, and 7, and those in the 
elements of the orbit derived from them are generally of the same 
order as that of the terms omitted in the expressions for 7, and 7,. 
Accurate and at the same time simple expressions for 7, and 7, 
have been given by J. W. Grpss’). 
The purpose of this paper is to develop, according to Gress’ method, 
expressions for n, and 7, which include the terms of the 4 order; 
at the same time a new derivation of Gripes’ relations is given. 
In the ellipse sought let P be the position of the heavenly body 
at the time ¢, # and y its heliocentric rectangular coordinates in the 
DJ. W. Gises: On the determination of elliptic orbits from three complete 
observations. Memoirs of the national academy of sciences. Vol. IV, 2; p. 81. 
Washington 1889. 
