Chase.] 296 |Jan. 18, 
corresponding harmonic quotients, the greatest deviation being less than 
one per cent., that they may be properly regarded as illustrations of second- 
ary interferences ; introducing two harmonic triplets, with a uniform de- 
nominator difference of 2 a, (n+2a4,4a,6a;n+ 6a, 8a, 10@). 
The bracketed divisors indicate tertiary harmonics, based on denominator 
differences of a/ = .0153: 1.1530 = 1+ 10a’; 1.4487 = 1 + 29a’; 1.7650 
=1-+ 50a’; 1.9180 =1-+ 60 a. The greatest difference between the 
theoretical and observed values is less than 3 of one per cent.; the other 
differences range between ?; and ;, of one per cent. 
Among the subordinate spectral lines there are some as I have shown 
elsewhere,* which are closely represented by the denominators 7 + 2 a, 
nm+4an+5a,n+74a, n+8a, n+9a. But, on account of the 
great number of faint lines, such accordances are less satisfactory than 
those which can be found in the lines which are more widely separated and 
more prominent. 
In planetary aggregation the interference waves have manifested their 
influence most strikingly at luminous internodes. The denominators 
are exponential, indicating roots which are to be extracted, instead of 
divisions which are to be made. It will be noticed that the first six expo- 
nential denominators in the following table, are arithmetical means between 
the adjacent numbers in the primitive series of nodal divisors in the fore- 
going table, and that the others are formed by successive denominator in- 
crements of } a. 
Exponential 
Denominators. Roots. Observed. 
1.0000 6453 64538 Neptune. 
1.0536 4130 4122 Uranus. 
1.1530 2015 2050 Saturn. 
1.2448 1150 1118 Jupiter. 
1.3366 708 728 Freia. 
1.4284 465 473 Flora. 
1.5202 321 32% Mars, 
1.6350 214 215 Earth. 
1.7497 150 155 Venus. 
1,8644 ital 110 Ven.-Mer. 
1.9792 : 84 83 Mercury. 
2.0939 66 64 Mercury, s. p. 
2.2089 53 538 Mercury, c. o. 
” 
The ‘‘observec values are the mean planetary vector-radii, in units 
of Sun’s radius. ‘‘ Ven.-Mer.’’ is the arithmetical mean between Venus’s 
mean distance (155) and Mercury’s secular perihelion (64). ‘‘ Mercury, 
c. 0.’ is the centre of spherical oscillation (,/ .4 ) of a nebula extending 
to Mercury’s mean distance. 
The harmonic interferences in the spectra of chemical elements may 
probably be best studied, by beginning with those which contain few 
* Ante, p. 110. 
