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three columns we could form linear equations to find small corrections to the mean radii of the three shells. But the improvement would be of the nature of a compromise, and would involve

(a) A study of the accuracy of our knowledge of refractions at small altitudes. We should have to settle, for instance, what error at 89° Z.D. can be made to obtain an improvement of 1" at Z.D. 80°.

(b) A more accurate computation of the differences for 10'. Those given above were found in the way made clear by the following example :

log cos 18' log cos 10° 9′9933455 = log cos (10° + 16′0)

log cos 29′ log cos 10° = 9′9933361 = log cos (10° +41′3)

log cos 40' log cos 10° 9′9933221 = log cos (10° +79′0)

I =

Ist 2nd diff. diff.

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The first two lines are formed to find the refraction due to the inmost layer (with constants 18' and 29') for altitude 10°, the result being 25" 3 as given in the 2nd column. By adding the 3rd line we find that the refraction of a shell with constants 29' and 40' (each 11' greater than the former) is 37"7; so that the difference for 11' is 12"4; and for 10' would be 11"3, which is given in the 7th column. But in forming differences of differences slight inaccuracies are multiplied, and inspection of the figures near the top of the last three columns shows that (since the numbers are nearly equal for all three shells) we are really concerned with differences of these again.

I

For the present, no attempt will be made to improve the approximation, and we turn to some other points.

25. It is of some interest to deduce the heights and refractive indices of the shells above found empirically.

One Shell.

log (1 + z) = log sec a=0'0002357 = log 1'000544

log μ= log cos Blog cos a = o ̊0001174=log 1'000270. The "height of the homogeneous atmosphere" is thus indicated as 4000 x 000544 miles, or 2.18 miles, which is a good deal smaller than that assigned by total pressure.

This is not surprising, for in replacing a series of successive bounding surfaces by a single surface, we should expect the equivalent single surface to lie in the midst of the constituents. We can imagine, for instance, a uniform graduation which would bring the equivalent surface to the mean height, i.e. would give us a "height of the homogeneous atmosphere" just half that assigned by calculating the total pressure; and it would not be surprising to find that the equivalent surface should be lower than this.

Two Shells.

Equations (4) of § 5 may be put in the form.

Log.

log (1+1) = log μ11+log sec a1 =0'0000309 log μ1/μ2 = log (1 + 1) + log cos ẞ1 = 0·0000199 log (1) log + log sec a=0'0005439 log μ1/μ=log (1 +%2) + log cos ẞ2 = 0·0001174 If we put μg 1, then μ1 = 1000270, μ2 = 1'000224. of the bounding surfaces in miles are z1 = 0.28,

=

%25'00.

Three Shells.

Number.

1'000071

1'000046

1'001253

1'000270

The heights

The method of formation of the quantities has been made clear

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It does not seem likely that these figures can have any physical interpretation. They merely emphasise the fact that a serious part of the refraction takes place near the Earth, and not high in the atmosphere. By comparing the columns for the separate shells shown in Tables I. and V. it will be seen how rapid is the increase in importance of the inner shells as we approach the horizon. In Table V., for instance, the two inner shells together only give half the effect of the outmost at moderate Z.D.'s; and they do not produce so great an effect as the outmost until we reach Z.D. 88°, when they surpass and ultimately double it in the remaining 21 or 3°. This suggests that any influence of meteorological phenomena on astronomical refractions must be sought at large Z.D.'s, which is in accordance with experience. But discussion of such points in detail cannot be undertaken at present.

SS1-2. Introductory.

Summary.

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$$ 3-6. A graphical representation of the refraction of a homogeneous spherical shell of atmosphere and of several such shells.

$$ 7-8. Condition that two or more such shells should be replaceable by an equivalent single shell.

SS 10-15. Empirical determination of constants for a single shell from observed refractions at Z.D. 45° and at the horizon. § 16. Extension of formulæ to two or more shells.

§ 17. Numerical results for one shell. The errors at 80°, 85°, and 89° are respectively 6", - 40", -281". The hypothesis cannot be said to fit the facts at all beyond Z.D. 60° or 70° at most.

See Table I.

SS 18-22. Numerical results for two shells. See Table I. The errors at 80°, 85°, and 89° are reduced to -1"5, - 7′′, and +2”—a very fair accordance for so obviously rough an assumption.

S$ 23-24. Numerical results for three shells: see Table V. The improvement is not great, much less marked than before; and it seems clear that the principal step was taken in passing, from one shell to two.

§ 25. Some numerical values for heights, etc.

Fourth Paper.

The Perturbations of Halley's Comet in the Past.
The period 760 to 1066. By P. H. Cowell, M.A., F.R.S., and
A. C. D. Crommelin, B.A.

We have again to acknowledge the kind assistance of Dr. Smart and Messrs. F. R. Cripps and Thos. Wright in these calculations.

In the last paper, M.N., lxviii. 5, p. 378, we found March 27 for the approximate date of perihelion passage in 1066, and 44" 686 for the value of n at that date. For the preceding passage we used the date given by Hind (989 September 12) for the purpose of computing the perturbations, and the following results indicate that this date is correct within a few days:

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For this and earlier revolutions we neglect the Neptune perturbations as trifling, and take those for Venus, Earth, and Uranus from curves constructed from the results already calculated; in the case of Venus and the Earth these curves are for the combined results of the mechanical quadratures and the definite integral. Needless to say, we do not claim that the results from these curves are absolutely accurate, but such accuracy is uncalled for, since our calculated Jupiter and Saturn results are liable to sensible errors through the uncertainty attaching to our assumed position of the comet's orbit plane, etc. The resulting value of n at 989 is 44" 686 + 1" 320 = 46" 006. And calculated period in days= 1296000-11155=27928. Now 1066 March 27=J.D. 2110500; 46'006 hence the calculated J.D. of previous passage is 2082572=989 October 9. This is 27 days later than Hind's date, a quantity sufficiently small to confirm his identification. We can reduce the discordance slightly by altering our assumed dates of perihelion passage, since the observations are not precise enough to fix the exact day. If we take them as 989 September 15 and 1066 March 25, we obtain from the revolution 989-1066 the values of n, 989 45" 969, 1066 44" 649; from the following revolution we obtain 1066 44" 688, 1145 44"*920.

The mean value for 1066 is 44"-668, to which corresponds 45"*988 in 989.

Proceeding to the revolution 912-989 we first took Hind's date, 912 April 1, and we give the perturbations deduced from this assumption:

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Taking 45" 988 as the value of n in 989, that in 912 is

45" 652, and calculated period in days=

1296000 - 9498

45652

= = 28181.

This gives J.D. 2054367 for the passage in 912=912 July 20.

This is nearly 4 months later than Hind's date, a larger quantity than is likely to arise from error in our calculation, and it is to be noticed that Hind's identification is antecedently very doubtful, being based only on observations on May 13, 15; we conclude that it is erroneous. There are vague references to other comets in 912, but nothing sufficiently precise to serve for identification. This is the first passage, reckoning back from the present time, that cannot be certainly identified with an observed comet.

Taking the calculated date 912 July 20, we obtain the following modified perturbations, 912 to 989. The value of [dʊ may be

taken the same as before.

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1296000" - 8691"
45"682

==

= 28180

This gives for the calculated period

days, practically the same as before.

Hence we take the date 912 July 19 as approximately correct for the perihelion passage, this result being from calculation alone, not from observation. As a check on its accuracy, we proceed to the revolution 837-912; a preliminary calculation indicates 837 February 28 for the preceding passage. It will be remembered that there is some question whether the numerous accounts of the brilliant object (or objects) that appeared in the spring of that year relate to one or to two comets. The date of perihelion passage of the earlier one, as investigated by Pingré, is indicated with tolerable precision as 837 March 1, the agreement with our calculated date being thus perfect. The other elements are also accordant except the position of the node, and it is well known that the indications of latitude in the Chinese accounts are somewhat vague, so that this is not a serious objection to the identification, especially as the comet was near the ecliptic throughout the

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