( 753 ) 
plane of the orbit, and 7— ZP, then « and y satisfy the following 
differential equations 
a v dy y 
as ei ——_=ij, 
rae r Gun r 
wherein we have put rk (t—t,) as independent variable instead 
of the time 4; rt is therefore the time reckoned from the epoch of 
the first observation and expressed in the unit for which, in the 
solar system, the acceleration —1 at a distance from the sun 
which is adopted as unit of length; # is the constant of Gauss 
[log k: = 8.235 581 4414 — 10). 
While designating the rectangular coordinates of PPP, by corre- 
x $ : EEEN ae 
: ant EY UT triangle P, ZP 
sponding indices I remark that n= ————- > ——————_—_ 
LY 3—Y 125 triangle P,ZP, 
satisfies a similar differential equation as a and y, namely: 
dn n 
—_- == — — =? 
dt? Ps 
At the times (v= Oi (ei 0,);- & CS) 
the values of 7 are 0 +n, + 1 and 
Sar Ns 1 
the values of 7 0 = —— 
ia ihe 
Consequently in the development of # in a series of ascending 
powers of t after Mac Laurin, the terms of the power zero and 2 
will be wanting. If in this expansion we do not go farther than the 
4h power of t, we require only 3 indefinite coefficients which may 
be eliminated from the following 4 relations: 
ns a eee eee L 
l= Kr, zie K,r," sie Kr oR IA 
a min Oan weet fe 
es IOA Iers 
The remaining relation yields an expression for », in 7,, T,, 7,7 
and the remainders /,, F, f, and F,. 
The indices which I have used for the remainders, indicate the 
order of these terms with respect to +; /,, for instance, which 
begins with Kr,’ is of the 4' order of tr, which is evident when 
we express the coefficients A in terms of the derivatives of for 
«=O and develop the latter by means of the differential equation 
for 2 as products of 7 
development : 
3 
» For clearness I shall here give this 
