1877. | ina [Chase. 
account of its novel application, it may be well to give it in full. The 
common astronomical unit is Earth’s mean radius vector; its value, in units 
of solar radius, is 214.86. The harmonic exponential numerator, is Nep- 
tune’s mean radius vector, which is 30.03386 astronomical units, or 
30.03386 x 214.86 = 6453.06 solar radii. The logarithm of 6453.06 is 
3.809766 ; log. log. 6453.06 = log. 3.809766 = 580897. By the same 
method we find log. log. Uranus = .558210 ; .580897 — .558210 = .022687 
= log. 1.0536. Uranus’s mean radius vector represents, therefore, the 
1.0536th root of Neptune’s mean radius vector, and 1.0536 is the denomina- 
tor of the first planetary fractional exponent. The first mid-nodal denom- 
inator, in the foregoing spectral-line series, between A -~ 1 and A = 
(nm + a) is (1 + 1.1068) + 2 = 1.0534 ; the second mid-nodal denominator 
is(n + a+ n-+ 2a) + 2 = 1.1527; and so on, until we reach the sixth 
denominator, when, perhaps on account of great nebular condensation, the 
harmonic denominator-differences become 2 of .0918, instead of .0918, 
bringing a second exact correspondence between the spectral and planetary 
denominators in the orbit of Venus. The following table contains all the 
figures that are required for the whole calculation : 
eapont Ae en 8+ og. 7.1. Log. r. II. Theoretical. Observed. 
1.0534 022593 558304 3.61689 1.28473 19.265 19.184 
1.1527 .062716 .019181 3.30576 .97360 9.410 9.539 
1.2445 094994 485903 3.06128 02912 5.359 5.427 
1.5368 126066 454831 2.84991 O1LT75 3.294 ? 
1.4281 154758 .426139 2.66771 .30000 2.165 2 
1.5199 .181815 399082 2.50658 17442 1.494 1.524 
1.6346 213412 .867485 2.33070 7.99854 299% 1.000 
1.7494 .242889 .838008 2.17775 7.84559 TOL .698 
1.8641 .270469 .310428 2.043875 T.71159 O15 10 
1.9789 296424 284473 1.92519 7.59803 392 387 
The log. logs., in the third column, are obtained by subtracting the logs. 
of the exponential denominators (column 2) from the log. log. of the ex- 
ponential numerator (.580897). Column 4 contains the antilogs. of column 
3; column 5 is column 4 reduced to logs. of Earth’s mean radius-vector, 
by substracting log. 214.86 — 2.332155 ; column 6 contains the antilogs. 
of column 5. Column 7 gives the mean distances of Uranus, Saturn, 
Mars, Earth, and Mercury ; the mean aphelion of Jupiter ; the mean peri- 
helion of Venus; and the arithmetical mean between Mercury’s secular 
perihelion, and Venus’s mean distance. 
We are now prepared to find the significance of the remaining Fraun- 
hofer lines, which is shown in the following table : 
Line. Wave Length. Denominator. Planetary Den’rs. Theoretical Den’rs. 
C 656.67 1.1590 1.1576 == Sat. p.* ‘ 
E 527.38 1.4434 Asteroidal. 
b 517.70 1.4704 1.4740 = n + 5a 
G 431.08 1.7660 1.7640 = Ven. s.p. 
H 397.16 1.9166 1.9139 = Mer. a. 
* p., mean perihelion; Ss. p., secular perihelion; a., mean aphelion. 
