CORRESPONDENCE. 



ASTRONOMICAL OIF.RIES. 

 To the Editors of " KNOwi.KnGE." 



Sirs. — I. — The subject of the brightness of a surface is one 

 presenting various problems which it is impossible to discuss 

 here. Mr. Bartrum will find a few of these in Parkinson's 

 "Treatise on Optics," Chapters II and \. In estimating the 

 brightness of Venus, Godfray assumes that the illuminated 

 surface appears uniformly bright and of the same degree of 

 brightness in all aspects. .As this is not true, the results 

 obtained are only approximate. 



It is interesting to notice, although bearing only indirectly 

 on the subject, that the half Moon does not give half as much 

 light as the full Moon. The visible surface, except at full 

 Moon, is darkened by shadows cast by the irregularities, and 

 so the amount of light is diminished. 



II. — The major semi-axis is called the mean distance, and 

 nnist not be confounded with the average distance. This 

 Latter depends upon what is taken as the independent variable. 



The mean distance of a planet from the Sun can be easih' 

 found by taking the Arithmetical Mean of the greatest and 

 least distances. The former is a + ae, and the latter is 

 a — ae : the .Arithmetical Mean of these two is a, the Major 

 Semi-Axis. 



III. — The gradual decrease of the eccentricity of the F.arth's 

 orbit which has been going on for thousands of years, and 

 which w'ill continue for a long time yet. causes the mean 

 motion of the Moon to increase. It is difficult to explain this 

 in ordinary language, and a niathematical investigation is 

 necessary. 



Let S denote the mass of the Sun, K his distance from the 

 Rai'th, and r the distance between the Earth and the Moon. 

 When the three bodies are in a line the disturbing component 

 of the Sun upon the Moon is 



r 1 IT 



L(R-rl- R-J 

 where m is a constant. This is evident, since the disturbing 

 effect of the Sun upon the motion of the Moon is the difference 

 of its acceleration upon the Earth and Moon. 



Now r is verv small compared with R. so that the above 

 2/1 Sr 



rt2R-rn 

 R'lR-rF-J 



equation becomes D = 



R' 



The disturbing acceleration in any part of the orbit of the 

 Moon depends upon the distance of the Sun from the Earth, 

 varying inversely as the third power of this distance. Let us 

 find the average for a whole revolution of the Earth, which we 

 may take to be performed in a time T. say. Let D' be the 



4 ^ Sr ri; dt 



T y u ' R" 

 From Kepler's Law that the radius vector sweeps out equal 



2 ^' dt 

 unit time 



average disturbing effect .'. D" 



areas in equal times, we obtain 

 h say. 

 .'. R- de = h d t. 



area described in 



.'. D' 



4m Sr 

 Th 



The polar equation of a conic 

 b- 



the semi latus rectum 



.'. D' = 4m Sra 

 Th b- 



gives 



T (l-+e cos H) d e. 



Now/ il+e cos 0) d rt = «+e sin fl = 7r, taking w and o as limits 



.'. D' = 



4 M Sra T 

 Thb' 



4 M Sr TT 

 ThaTd-e') 



It is easily pro\cd that 



h" = M L C and T = 



2 7rai: 



VmC 



where C is a constant. (See Williamson 

 Dynamics, Chapter VII. Equations 30 and 3 

 proof. I 



.'. Th = 2^ail a/17=2 



and 



1, for 



D' 



\/a(l ■ 

 2MSr 



e-| 



Tarlton's 

 a simple 



a" (1 - e"l i 



It is easily seen that as e decreases, the denominator 

 increases, and therefore D' decreases. The efficiencv of the 

 Sun in decreasing the attraction of the Earth for the Moon 

 therefore decreases, and the mean motion of the Moon 

 increases. 



It should be said that the amount of variation calculated in 

 this way, about six seconds a century, does not agree with the 

 amount of acceleration obtained by comparing ancient with 

 modern eclipses. It has been supposed that the discrepancy 

 arose from a lengthening of the day, due to a retardation of 

 the E.arth's rotation from friction of the tides. The subject 

 is. however, too long to discuss here. 



iREV.I M. DAVIDSON. 



THE FOURTH DIMENSION. 

 To the Editors of " Knowledge. " 



Sirs. — 1 was greatly interested in Mr. .A. L. .Annison's 

 paper on " The Fourth Dimension " which appeared in the 

 June number of " Knowledge," and I have been expectantly 

 looking for the corollary to his useful piece of analytical 

 reasoning : I may have missed it, but I certainly have not 

 noticed it. 



It appears to me that the logical deduction from his series 

 of analogies is that the Fourth Dimension is Density (5) ; I 

 believe you will find that this fits in completely, and satis- 

 factorily answers the demands of a Fourth Dimension. 



Hence all physical substances could be expressed completely 

 in terms of L X B X H X 6. 



.As this Dimension seems to have been overlooked because 

 so well-known under the term of " weight," it appears to me 

 probable that other dimensions may perhaps be recognisable 

 as yet other well-known physical or chemical properties of 



'"'■^^^^^- ALFRED J. .MULLINS. 



COMETS 1911 a AND c. 

 To the Editors of " Knowledge." 



Sirs. — The following obser\ations of Wolf's and Brooks' 

 comets, respectively 1911 a and c. were made on the evening 

 ot August 20th, 1911, by the aid of the forty-inch refractor of 

 this observatory (Verkesl. 



Wolf's comet (periodic), was very apparent in the forty-inch 

 and was in a rich field. Small. Slight indication of a 

 nuclear condensation. Near a small star. 



Brooks' comet was observed, first with the four-inch finder 

 of the great telescope. The nuclear condensation was a little 

 more apparent than in previous evenings, but the comet was 

 generally about the same. The nucleus and matter extending 

 in all directions from it were all that were visible at one time, 

 on account of the size of the field of view ; the former, how- 

 ever, presented a fairly large disc and was nebulous in appear- 

 ance. The light of the nucleus was practically white. There 

 were several comparatively bright stars in the same field, 

 which were actually visible through the head of the comet, 

 apparently undergoing no loss in brilliancy whatsoever. The 

 field itself was very luminous, and in some parts the nebulo.sity 

 was more apparent than in others, w'hich fact indicated the 

 presence of a tail extending out into space, beyond the head 

 of the comet. FREDERICK C. LEONARD. 



411 



