812 
ark R 
1 7 d; 
ge Er EN NN 
er dw F 
ee 7 dt 
and this value must be subtituted in the formula (3). In doing 
this, we can either express the coordinates and velocities in the 
osculating elements, or the latter in the former, by the well known 
formulas 
ca 1 i 1 + ecos v dr __ ane sin v 
En eS 
ie i cor ha hal VAs ae 
dt 
We then find 
dn = — qd, @r a are Ae es Fs 
J a’n (1 + ecosv) 
or 
I\ 2 
On = 9S, (5) cos 8 Wer (6) 
Ze 
Where we have put 
3nym? (l—e) ah’, 
g = 
: 100a,a 
2 
We can with sufficient accuracy *) take in the formula (5) a,’n, = «°n, 
ND 
and in the formula (6)V1—e =V1—e,”. The formulas can, however, 
not be used for the computations, unless they are so developed as 
to contain only such quantities as can be easily derived from existing 
tables. 
tb The formula (6) is derived by Borritncer from the vis viva integral. In this 
derivation he introduces a couple of approximations, which are unnecessary, and 
which are the reason why the factor 1 1—e® does not appear in his formula. On 
his page 12 he takes tani for sini. If we retain sind and replace it by its value 
1d 
, the square root drops out of the formula, and consequently the approximation 
Vd 
introduced on page 13 in the development of this sa me root is also unnecessary. We 
En) 
8 ddr > EUS de c 
then find An = — —- J — . Now we have V 1—e? =r? ze and 7 =—. Borriin- 
an dt dt a 
BV 1—e? dx 
cer’s formula (1) on page 13 thus becomes An = J, and his formula 
ij dy 
(1) on page (18) then becomes identical to our formula (6). 
*) See however the footnote on p. 815. 
_— a 
