52 
PERIODIC ORBITS. 
[1 §15 
If at any stage the factor to be extracted becomes small, the whole row 
to which it belongs becomes large, and the symmetry may perhaps be 
seriously affected. In this case it is well not to choose this pair of centres 
of elimination, but to take another pair, leaving this pair to a later stage in 
the calculation. 
If the determinant is negative, a negative factor will be extracted at 
some stage. In all the cases which have been worked out it is easy to 
see that no other negative factor will ever arise, and thus the determinant 
will remain clearly negative. Most of the determinants have been written 
with 17 columns and rows; then beginning with (- 8, —8) and (8, 8) I find 
that it is often possible to erase 8 columns and 8 rows on a single sheet of 
paper, with scarcely any modification of the central part of the determinant. 
Thus the determinant which at first had 289 spaces (although many only 
contain zeros) is reduced to 81 spaces, with but little labour. 
The multiplications have been done with Crelle’s table, but a specially 
computed auxiliary table of products, from , 000 x ’000 up to '040 x ’040 to 
three places of decimals, has rendered the work much more rapid. 
I believe that the values obtained by this process are correct to within 
about one per cent. For the same determinant when reduced with different 
order of elimination agrees with its previous determination within less than 
that amount of discrepancy. 
Part II. 
§ 15. Periodic Orbits. 
An orbit in which the third body can continually revolve, so as always to 
present the same character relatively to the two other bodies, is said to be 
periodic. If the motion is referred to a plane which is carried round with 
Jove and revolves about the Sun as a centre, any re-entrant orbit of the 
third body is periodic. Periodic orbits may consist of any number of revolu 
tions round either of the primaries, or round other points in space. Periodic 
orbits, which are only re-entrant after several circuits, are much more 
difficult to discover than those which only make a single one; as hardly any 
thing is known up to the present time about this subject, I determined to 
confine my attention to “simple periodic orbits,” which are re-entrant after a 
single circuit. This definition of a simply periodic orbit must not preclude 
the consideration of orbits with loops, for the inclusion of such loops is 
necessary to the comprehension of the subject. 
It appears from the differential equations of motion that periodic orbits 
must in general be symmetrical with respect to the line of syzygy; or if any