Appendix

#### NOTE I

The critical velocity at the surfaces of the planets is found as follows: --

Using the usual symbols we have:

fdt = dv
therefore fds = vdv

And as f = -m divided by s squared, since the force tends to decrease the coordinates, this becomes -mds divided by s squared = vdv

Integrating:

m divided by s = one half times v squared plus c, of which the definite intergral from s1 to s2 is

m divided by s1 minus m divided by s2 = one half times v1 squared minus one half times v2 squared

Hence, since at infinity the velocity is 0, the equation for a fall to a planet's surface from infinity is

m divided by r = one half times v squared

r being the radius of the planet and v the velocity acquired at its surface from a fall from infinity, which is the same as the velocity needed for projection from its surface to infinity.

To find m we have in the case of the Earth p = 32 ft. a second at its surface; this gives us m in terms of g, that is, f. For the other planets we need only to introduce their masses and radii in terms of those of the Earth and then multiply the value f or the Earth by the square root of the ratio.

The result is that we find the critical velocity for the several planets and for the Sun to be as follows:--

```Mercury 2.2 miles a second (probable value).
Venus 6.6 " " " " "
Earth 6.9 " " "
Moon 1.5 " " "
Mars 3.1
" " "
Jupiter 37. " " " (mean value)
Saturn 22. " " " " "
Uranus 13. " " " " "
Neptune 14. " " " " "
Sun 382.
" " "```
While the probable maximum speed of the molecules of some of the common gases at 0 degrees Cent. are as follows: --
``` Hydrogen 7.4 miles a second
Water vapor 2.5 " " "
Nitrogen 2.0 " " "
Oxygen 1.8
" " "
Carbonic dioxide 1.6 " " "```

#### NOTE II

The change in the apparent size of the equatorial diameter as compared with the polar one as the phase increased, suggesting the unconscious measurement of a twilight upon the planet, becomes still more striking when, in addition to the October-November measures mentioned in the text, the measures from July to October are considered in connection with them. Tabulated chronologically, the whole are as follows: --
```MEANS
Polar Diameters
July (6 to 22 inc) 9.976 0".13 0° 9.933
Aug (11 to 21 inc) 9.362 0".04
0° 9.325
Sept (20 to Oct 5 inc) 9.401 0".012 0° 9.355
Oct (12 & 24 to 30 inc) 9.375 0".011 1° 9.336
Oct
(15 to 23 inc) 9.379 0".028 2°.5 9.339
Oct (12 & 24 to 30 inc) 9.375 0".028 1° 9.336
Nov (2 to 21 inc)
9.390 0".012 4° 9.350
July (6 to 22 inc) 9.691 0".11 46°.5 9.672
}9.680 }0".08
Aug (11 to 21 inc) 9.666
0".15 41°. 9.645
Sept (20 to Oct 5) 9.523 0".010 20°.5 9.490
Oct (12 & 24 to 30 inc) 9.457 0".016 7° 9.417
Oct (15 to 23 inc) 9.429 0".010 1° 9.385
Oct (12 & 24 to 30 inc) 9.457 0".016 7° 9.417
Nov (2 to 21 inc)
9.545 0".015 19° 9.514```
It will be seen that, except for the July value, the size of the polar diameter comes out essentially the same throughout. Now, during July the polar cap was very large, and covered the southern part of the disk at the point where the polar diameter was measured. As it was much brighter than the rest of the disk, its irradiation must have been correspondingly great, and this would have had the effect of increasing the apparent length of the polar diameter beyond its true value.

The equatorial measures, on the other hand, show a systematic increase as the phase increased; and they do this on both sides of opposition. The increase, it will be noticed, is much greater than the probable errors of observation.

By PanEris using Melati.

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