‘ 
D. Kirkwood on certain Harmonies of the Solar System. 11 
with considerable interest. Toward the close of the last century, 
-rofessor Bode, who had given the subject much attention, pub- 
lished the law of distances which bears his name, but which, as 
he acknowledged, is due to Professor Titius. According to this 
formula, the distances of the planets fronmMercury’s orbit form 
a geometrical series of which the ratio is two. In other words, 
if we reckon the distances of Venus, the earth, &c., jrom the 
orbit of Mercury, instead of from the sun, we find that—interpo- 
lating a term between Mars and Jupiter—the distance of any 
member of the system is very nearly half that of the next exte- 
rior. The series is usually expressed as follows :— 
4 = 
4+3x2°= 
44+3X2!=10 
44+3 x 22=16 
443 X 23=28 
&e. &e. 
The numbers 4, 7, 10, &., represent approximately the rela- 
tive distances of Mercury, Venus, the earth, &c., from the sun. 
The ninth term, however, which corresponds to Neptune, is 388, 
instead of 300. It was, moreover, remarked by Gauss that ‘the 
member which precedes 4 +3 should not be 4; i.e. 4+ 0, but 
4+14. Therefore, between 4 and 4+ 3, there should be an 
infinite number; or, as Wurm expresses it, for n=1, there is 
3.” 
a+ br", a+br*-!, a+ br”, &e. 
In the scheme of Bode and Titius, a=4, 6=8, and r=2; 
in that of Wurm, a= 887, 6 = 293, and r=2. Both fail to 
represent, even approximately, tlte relative distances of Mercury 
and Neptune. 
I have never doubted that the planetary distances were ar- 
ranged in some discoverable order. These failures, however, in 
the series of Titius have seemed a sufficient cause for its rejec- 
tion, or, at least, some considerable modification. I have, for 
many years, been devoting such thought and attention to the 
‘subject as circumstances would permit, and I now propose to 
submit my results to the public. 
In the American Journal of Science and Arts, for September, 
1852, the fact was noticed that, “if we commence with Neptune, 
most remote planet known, we shall find that the primary 
© Cambridge Philosophical Trans., vol. viii, p. 171. ; 
