ry 
236 
NLR RE 
[JANUARY 7, 1897 
This question of the discordance between theoretical and 
observed results has been recently treated by Dr. Hausdorff, and 
greater importance has been given to the memoir by a review 
from Dr. Kempf, ina recently-issued Verteyahrsschrift ; and this 
last, while traversing some of the views of the authors, is, on 
the whole, as satisfactory and reassuring as the views of Prof. 
Langley and Dr. Seeliger have been disturbing. Dr. Hausdorff, 
concerned to reconcile theory and observation, examines in the 
first place Laplace’s theory with great accuracy to trace whether 
the deviations are due to any incompleteness in the theory itself. 
The principal result of this inquiry is to adda term of rather 
complex character to the simple formula to which Laplace re- 
duced his expression for atmospheric absorption, but which 
avails nothing, either in actual practice or in offering a solution 
of the original difficulty. The new coefficients might be of use 
if their physical interpretation added anything to our knowledge 
of the atmosphere ; but whether this be assumed of infinite or 
limited extent, the whole effect of the new introductions, even 
in small altitucles, can be represented by an alteration in the 
constant. How little improvement, if any, the consideration of 
terms of a higher order has had on the computed quantities, is 
shown by the following table, in which Miiller’s observed values 
of atmospheric absorption (M.) (expressed in logarithms) are 
compared with those of Hausdorff (H.), and the more simple 
expressions of Laplace (L.). 
Zs Oh) (H.) (L.) (M)-(H.) (M.)-(L-) 
(o) 0 000 0"000 0'000 (o) (o) 
20 0004 CO5 ‘005 = I -— 1 
40 0'024 1023 024 + 1 fo) 
60 "092 070 ‘O77 +16 +15 
7o o'180 “146 147 +34 + 33 
75 0'261 215 216 +46 +45 
80 -0°394 "350 3352: | $44 42 
$2 0°47 “447 “448 +30 +2 
$4 0°607 “597 508 +10 + 9 
56 0'$46 850 855 = 16) =" (0) 
yp ibis) 1°209 1*200 — 33 —2 
Practically identical figures result whether the atmosphere be 
supposed infinite or of limited extent. 
The essential service that Dr. Hausdorff has rendered is to 
show that any considerable improvement in Laplace’s theory is 
not probable. Dr. Kempf now renders a still greater service by 
showing that any improvement is not needed. He raises the 
question whether these observed discrepancies are not rather 
due to observation, and removable by more appropriate methods 
of discussing the observation. Dr. Kempf remarks that the only 
observations available for examination are those of Dr. Miiller. 
whether made at Potsdam or on the Santis. Seidel is brushed 
aside with a scanty reference. Pickering and Pritchard do not 
get even this recognition. It would probably be objected to the 
latter that he had not observed below 75° Z.D., and it is only 
after this altitude is passed, that the deviations between observation 
and theory become of noticeable amount. But Prof. Pickering’s 
observations of circumpolar stars at upper and lower culmina- 
tions seem to be available, and should not be rejected without 
reason or excuse. But it may be urged that the real object of 
the inquiry is to explain Dr. Miiller’s observations, and from the 
connection that has long existed between Drs. Kempf and Miiller, 
the remarks of the former become the more valuable, since it may 
be assumed that Dr. Miiller is cognisant of the treatment that 
his observations have received at the hands of his coadjutor, and 
has tacitly acquiesced in the process. 
Dr. Hausdorff has declined to use the Santis observations, 
-because the corrections for light extinction are founded on 
Laplace’s theory ; but Dr. Kempf shows how the observations 
can easily be made to furnish results independent of any theory, 
and consequently the discrepancies between the observations and 
Laplace’s values can be easily exhibited. Expressed as light 
ratios in logs., the unit corresponding to the third place of 
decimals, the differences Miiller-Laplace are shown in the 
following little table, in which no regular systematic progress 
is noticeable. 
Z.D. (M.-L.) Z.D; (M.-L. 
18°6 —10 62°8 ae er a 
29°3 + 6 (J ae GPR HEM) 
37°2 - 5 79°5 meee) Cates Se 5) 
44°6 te) $2°9 SEierce eye (2) 
57°6 apt) 858 = a 
NO. 1 41 9, VOL. 55 | 
Not only are the différences small, but in no case do they 
exceed twice the probable error derived from the discussion of 
the observations themselves. Encouraged by this satisfactory 
agreement, Dr. Kempf applies himself to the Potsdam results, 
and recalls the fact that the extinction table, on which Dr. 
Hausdorff so much depends, is really the result of the combination 
of two separate processes of observation. The first part extends 
from the zenith to an altitude of 10°, and has been derived by 
comparing the light of Polaris with that of five different stars 
at every possible Z.D. between 0° and 80°. For greater Z.D., 
however, another series has been obtained by observing, on very 
clear nights, the differences of lustre of objects at their rising or 
setting up to some 10° of altitude, and deriving from the differ- 
ences the amount of absorption at the various Z D.s. Sucha 
break in the continuity is perhaps regretable, but to some extent 
unavoidable. Stars that are visible in the horizon do not reach 
the zenith of Potsdam within some 15, and, of course, stars 
culminating near the zenith do not approach near enough to the 
horizon. Comparing each part of Miiller’s general extinction 
table with Laplace’s theory, Dr. Kempf obtains the following 
result (M.-L.) I. ; but, for very sufficient reasons, on discarding 
the last observation at 80° Z.D., which is evidently discordant, 
and affects the accuracy of the determination of the general 
coefficient of transmission, he obtains the second series marked 
(M.-L.) II., in which the agreement leaves very little to be 
desired. 2? 
Z.D. (M.-L) I. (M.-L.) IT. Z.D. (M.-L.) I. (M.-L.) II. 
OS) ofa ae -2 2°5 +10 +5 
BO es) = 3 =a 75 ar ae) +2 
BOM) 2 -3 775 or =i 
GONG. .-h3 fo) 80 -17 — 
79 9 ta) 
Nearly as satisfactory is the comparison of the theory with the 
second part of the table, that below 80° Z.D. The greatest 
deviations, when expressed in magnitude, do not exceed 
0°025 m., and any one acquainted with the difficulties attending 
photometric observations so close to the horizon, will rather be 
surprised that the agreement is so close, than tempted to cavil 
over the small discordances. In the first part, the coefficient ot 
transmission is 0°81 ; in the second, 0°85: the two parts, there- 
fore, correspond to different degrees of atmospheric transparency, 
and they cannot be represented by one and the same curve. 
The explanation offered by Dr. Kempf will probably command 
a general assent, and it will be admitted that he has made out 
his case, that Laplace’s theory of: absorption corresponds 
exactly to the actual conditions within the limits of accuracy at 
present attainable. 
If Dr. Kempf has provided the true explanation, it is of 
little use to follow Dr. Ilausdorff in his further investigations, 
based as they are on the entire conviction that Miiller’s 
extinction table is exact, and that the discordance between theory 
and observation is real. His attempts to devise new formulz 
on various hypotheses are not very satisfactory, simply regarded 
as interpolation results; and his failure to represent Dr. Miller’s 
figures more closely, tends to confirm the probability of Dr. 
Kempf’s suggestion. Of course, by the extension of formule 
to an inconvenient length, and the introduction of a sufficiently 
large number of unknowns, derived from the observations them- 
selves, a close agreement can be forced ; but even then one may 
be driven to such inconvenient consequences as that the intensity’ 
of light at approximately the sea-level is greater than that at the 
top of a mountain 2500 m. high. On no supposition (and in 
some instances the ingenuity displayed in the construction of 
hypotheses is considerable) can a formula be found that more 
closely represents observation than does Laplace’s; and though 
the author did not propose to himself to establish this fact, he 
has rendered no slight service to theoretical photometry by the 
practical confirmation his work affords. 
W. E. PLUMMER. 
ON CERTAIN VESTIGIAL CHARACTERS 
IN MAN. 
>EEING that Prof. Huxley, with his well-known candour, felt 
constrained to admit that the study of rudimentary or vest- 
igial characters had done more than that of any other class of 
facts to produce general acceptance of the doctrine of evolution, 
and that at the same time he acknowledged the double-edged 
