( 755 ) 
Re Seater oe T; \ F “Eats Te) 
Jennie 12 as 5 | 4 in 2 12 
R = ciel 
d ‘le Pie CN Pie 
[rs 
This remainder is apparently of the 4° order with respect to the 
intervals. If we neglect the terms of higher order than the fourth 
we can replace in R,: f, by Art”, F, by Kr”, f, by 20K,r,* and 
F, by 20K,r,° ; and we obtain as supplementary term, accurate to the 
fourth order 
1 
== a IE (es 5e T;) (tr, a T;) (27, Fri T;) (tr, a 2r,) ’ 
J 
which expression vanishes on account of the last factor, in the case 
of equal intervals. 
The corresponding approximate formula for 7, can be derived by 
: … triangle PZP. ke ; : 
developing the relation ——~——___., depending on the time, in ascend- 
triangle P,ZP, 
ing powers of &(t, — tf) and further by proceeding in the same manner 
as we have done for 7,. The result for », is derived from the pre- 
ceding result by interchanging the indices 1 and 3, in which case 
t, stands for & (¢, — ¢,), hence : 
pee ipa Cre: 
Es 
Ga 12r, x 
I= Ea = R, 
T, 1 T, ar An T, 
Dn 3 
12r, 
The remainder of 7, is not only of the same order as that of 7,, 
but even in the 4% order it has the same absolute value, with a diffe- 
rent sign however. This appears clearly when, using the relation 
TT Ht, we express the correction of the 4% order for 7, in 
terms of t, and r,; this correction takes the following form, which is 
symmetrical with respect to rt, and r,: 
|E 
3 Kier, (ar vole, Hat) (EE) 
In the remainder of n, the coefficient L, may be assumed equal 
to A. Therefore these approximate formulae always give for 
n, FN, an accurate value (comp. p. 758), including the terms of 
the 4% order of the interval. 
The denominators of these expressions for 7, and 7,, although here 
different in form, are indeed identical; the expressions themselves 
agree with those derived from the fundamental equation adopted by 
GisBs between the 3 vectors ZP,, ZP, and ZP, which can be easily 
3 
