A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae
Proceedings of the National Academy of Sciences, 1929 · paper · Physics
Re-examines the relation between distances and radial velocities of extra-galactic nebulae, using only distances believed fairly reliable, estimated from stars involved in them and from mean luminosities in clusters. Motivated by a puzzling variable solar-motion term in earlier determinations.
Checked: PDF from europepmc.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance 9-group solution on p. 3 prints X = +3, Y = +230, Z = −133 but V0 = "247 km./sec." and D = "+ 33°". The printed components give V0 ≈ 266 km/sec and D ≈ 30°.
- p. 3 solution table, 9 GROUPS column
247 km./sec.
Only the authors can say what is right here.
Judge's note: I checked the solution table in the crop and on page 3. Taking the apex as −(X,Y,Z), the 24-object column agrees with itself: sqrt(65²+226²+195²) = sqrt(93326) ≈ 305.5, printed 306; A = 286°; D = atan(195/235.2) ≈ 39.7°, printed 40°. For the 9-group column, A = 269° matches atan2(−230,−3) ≈ 269.3°. But sqrt(3²+230²+133²) = sqrt(70598) ≈ 265.7, not the printed 247, and atan(133/230) ≈ 30.0°, not the printed 33°. No text in the paper explains a different weighting for these derived values. With V0 = 247 and D = 33°, Z ≈ 134.5 agrees with the printed Z, but the horizontal component would be about 207, not about 230. So either Y or the derived values may be misprinted, and only the author can say which. The arithmetic below compares squares, because the expressions cannot use square roots.
Second check: another model agreed
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Wording Page 172 says the 22-nebula mean point, "745 km./sec. for a distance of 1.4 X 10^6 parsecs, falls between the two previous solutions and indicates a value for K of 530". The two solutions give K = +465 and +513, so the implied K of about 530 lies above both rather than between them.
- p. 5 First paragraph
falls between the two previous solutions and indicates
- p. 3 Solution table
+465 50 +513 60 km./sec. per 106 parsecs.
Only the authors can say what is right here.
Judge's note: Compared the page 172 text with the solution table on page 170. The implied K exceeds both solution values, and at r = 1.4 both solution predictions fall below 745. The paper gives nothing that reconciles the word "between" with these numbers, so only the author can say what was intended.
Second check: another model agreed
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4 minor (typos and the like; most checked by the judge only)
Reference or label The Figure 1 velocity-axis ticks read "+1000 KM" and "500 KM" rather than km/sec, although the axis is labelled VELOCITY and the text uses km./sec. throughout.
- p. 5 Figure 1 vertical axis
500KM
Judge's note: Viewed Figure 1 on page 5. The tick labels omit the time unit. This is evidently shorthand, and the axis title VELOCITY makes the meaning clear, so it is a minor label point.
Second check: another model agreed
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Wording Table 2, NGC 3115: v = +600, v_s = +105 gives (v − v_s)/500 = 0.99, but r is printed as 1.00; every other row reproduces r exactly from (v − v_s)/500.
- p. 4 Table 2, NGC 3115 row
+105
Judge's note: Checked all 21 rows of Table 2 on the page 4 image. Only NGC 3115 departs, by 0.01. This is probably a rounding of v_s to 5 km/s. M_t = −15.5 rounds the same either way, so nothing downstream changes.
Second check: another model agreed
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Wording Text on p. 5 gives "the ranges, 4.9 and 5.0 mag." (same order as the means −15.3 for Table 2 and −15.5 for Table 1), but Table 2's M_t values run from 12.8 (N.G.C. 1700) to 17.6 (N.G.C. 4594), a range of 4.8.
- p. 5 Text after Figure 1
the ranges, 4.9 and 5.0 mag.
- p. 4 Table 2, M_t column
17.6
- p. 4 Table 2, M_t column
1700 ... 12.8; 4594 ... 17.6
Only the authors can say what is right here.
Judge's note: I read the Table 2 M_t column on the page 4 image. Its extremes are 12.8 (N.G.C. 1700) and 17.6 (N.G.C. 4594), and N.G.C. 404 has no entry. Table 1 on page 2 runs from 12.7 (6822) to 17.7 (4472), which gives 5.0 and matches the text. The order of the means (−15.3, −15.5) ties 4.9 to Table 2. Recomputing from the tabulated m_t and r (r = corrected velocity/500: 1115/500 = 2.23 and 580/500 = 1.16) gives about −17.64 and −12.82, a range of about 4.82, so rounding does not obviously give 4.9. I found no other set of values in the text. The discrepancy is 0.1 mag in a range quoted for a qualitative comparison, so I rate it minor and leave it as a query to the author.
Second check: another model agreed
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Typo "supports the validity of the velocity-distance relation in a very evident matter" should read "manner".
- p. 6 First line
evident matter.
Judge's note: The page 173 image shows "evident matter." as printed. This is a typesetting slip.
Second check: none (judge only)