CORRESPONDENCE. 



ASTRONOMICAL 

 7"o the Editors of 



QUERIES. 



■ Knowledge." 



Sirs, — In the course of Astronomical study I have lately 

 found difficulty on three points. I am venturing to lay these 

 points before your readers, in the hope that a solution may be 

 found me through the niediuni of vour columns. 



Let a denote tl 



Figure I. 



?nii-niajor axis, A O. 

 .. minor .. C O, 



Ihpticity 



Then R 



AO 



r „ ,, variable radius vector, P F, 



T ,, „ complete period of revolution, 



S „ ,, perimeter of the ellipse, that is, the rectified 



complete path of the body, 

 R ,, „ arithmetic mean of the greatest and least 



values of ;-, 

 Rf ,, „ time-average value of ;-, 

 Ra t, ,. angle-average ,, „ 

 Ro M .. orbit-average „ „ 



BF+AF_RO+FO+AO-FO_ 



'' Vi''- 



-"[ 



1 + '-^' 



R..=. 



R., 



de = b. 



- ds = a. 

 S 



I. — A uniformly illuminated surface, \vhate\er be its form 

 and whatever be the angle made by the line of sight to its 

 component parts, will appear uniformly illuminated where- 

 ever the eye be placed ; the amount of light that reaches the 

 eye will be proportional to, and depend only upon, the intensity 

 of the illumination and the solid angle subtended by the 

 boundaries of the surface at the eye. A familiar illustration 

 of this is the case of an opal gas globe illuminated by a gas 

 flame at its centre. The globe appears as a fiat uniformly 

 illuminated disc from all positions. This phenomenon is well- 

 known and readily explained. In the case, however, of a 

 surface illuminated by a point-source of light, and shining by 

 reflection, the intensity of illumination varies with the angle 

 of incidence, being, in fact, proportional to the cosine of that 

 angle. Considering the case of a sphere, the intensity falls off 

 to zero at the terminator of the illuminated portion. To an 

 observer, then, such a sphere does not show the same intensity 



at the different portions of its surface ; and the amount of light 

 that reaches the eye depends not only on the solid angle the 

 illuminated surface subtends at the eye. but also on the 

 varying intensity of illumination, due to the varying angle of 

 incidence of the light. 



Now the Nautical Almanac, in calculating the time of 

 greatest brilliancy of Venus, assumes that the illuminated 

 portion of the planet visible from the earth is of uniform 

 intensity, and that the brilliancy varies only with the solid 

 angle subtended at the earth. Godfray in his " Treatise on 

 Astronomy " gives the same method of calculation. Why 

 should an approximately accurate result be expected on such 

 an assumption ? 



II. — What is the usual interpretation of the expression 

 '■ mean distance " as applied to the radius vector of a body 

 revolving in an elliptical orbit about its primary ? 



The following interpretations occur to one, and there are 

 doubtless others. 



(a) The arithmetic mean of the greatest and the least value 

 of the radius \ector, which has the value a (see Figure 1). 



(6) The average value at eijual intervals of time which is 

 a (l+e';2l. 



(c) The average value at equal increments of the angle 

 made by the radius vector, which is /). 



id) The average value of the radius vector at equal intervals 

 of path of the body in its orbit, which is a. 



One reads frequently of the '" mean distance " without 

 qualification ; its value is generally taken to be the semi- 

 major axis ; but the time-average value would seem to be an 

 important interpretation, and this is not the semi-major axis. 



III. — It is recognised that the secular acceleration of the 

 moon's mean motion is partly accounted for by change in the 

 perturbation of the moon by the sun as the earth's orbit 

 slowly becomes less elliptical. Here is a difficulty I cannot 

 solve. I will endeavour to state it. 



With decrease of ellipticity the time-average value of the 

 earth's radius vector, a d+c" 2). grows less. As the radius 

 \ector decreases the sun's perturbing influence on the moon 

 increases. With increased perturbance the moon's motion is 

 retarded. But the moon is known to be accelerated. There 

 is then some fiaw in the above reasoning which I cannot 

 find. I crave help from one of your astronomical readers. 



Hampstead, X.W. C. O. BARTRCM. 



THE FOURTH DIMENSION. 

 To the Editors uf " Knowledge." 



Sirs, — I was very greatly interested in Mr. A. L. Annison's 

 paper on "The Fourth Dimension " appearing in your issue 

 for June, 1911. The writer, I think, very clearly shows that 

 there is no valid reason why the '" fourth dimension " should 

 not actually exist : but, it may be asked, does this amount to 

 a proof that the "' fourth dimension " does actually exist ? 

 1 hardly think that we may answer, yes ; and it is for this 

 reason that I venture to write to you, sirs, in order to call 

 attention to my own work upon the subject, because I believe 

 that I have been able to demonstrate, assuming the truth 

 (1) of the Euclidean conception of space, (2) of the principle 

 of the continuity of mathematical law, that the fourth and 

 higher dimensions do actually exist ; the existence of a third 

 dimension of space implying that of a fourth, and so on, to 

 infinity. As to the first of the above postulates (the truth of 

 the Euclidean conception of space), all our experience supports 

 it ; but at the same time, I believe the wording of my proof 

 might be modified so as to apply in the cases of the elliptic 

 and hyperbolic space-hypotheses as well as the parabolic or 

 Euclidean. As to the second postulate, this is altogether in 

 accord with all our experience, and is. therefore, rightly 

 employed in discovering facts which lie without the same. 



A very brief statement of my proof, which was originally 



311 



