June 1, 1883.] 



♦ KNOWLEDGE ♦ 



HAo 



of the distance A B exceeds the square of the distance of 

 A' B'. The same is true for all parts of the surface S S'. 

 Hence the illumination of A B is to the illumination of 

 A' B' inversely as the square of the distance of A B is to 

 the square of the distance A B. The same reasoning would 

 obviously apply if A B were inclined at any anjjle to P. 

 and the plane A' B' parallel to the plane A B. Hence wo 

 have the law that the illumiiMlioii of siini!af!i/ jtJaccd 

 opaque surfaces varies inverse!)/ as the square of the distance 

 cf the source of light. 



Next, as to the effect of the inclination of the opaque 

 surface to the light rays falling upon it. 



If A B C D, Fig. -4, represent the same areas as A B, 

 C D in Fig. 3, and A B be tilted to the position a h ; 

 then if the oval c </ be the intersection of the plane a b 

 with the conical pencil C P D, the area of the oval c d is 

 to the area of the circle C D (by a well-known property of 

 projection) as 1 to the cosine of the angle lietween the 

 planes A B, nr 6 (the point P being regarded as so far oil" 

 that c C and D d are appreciably parallel). But the angle 



between the planes A B and a 6 is the complement of the 

 angle between P c and the plane of a b. Hence the area 

 •c rf is to the area C D as 1 to the sine of the angle between 

 P c and the plane a b. Since then only the same amount 

 of light falls on the larger area c d as on the smaller area 

 D, it follows that the illumination of c d is to the 

 illumination of C D in the ratio of the sine of the angle 

 between P c and the plane a b. This being true for all 

 points of S S', we deduce the law that the illumination of 

 an inclined opaque surface varies as the sine of the angle 

 between the light rags and the surface, or as the cosine of the 

 angle between the light rags and a line perpendicidar to the 

 surface, this last-named angle being called the angle of 

 incidence. 



These laws are sufficient for the cases of the opaque 

 celestial bodies. For terrestrial olyects under illumination 

 from large sources of light — as, for instance, a body illumi- 

 minated Viy the sky, either the complete hemisphere of sky 

 {the sun biing supposed to be out of view) or some given 

 portion thereof — other considerations have to be taken into 

 account. For lines from different parts of the sky do not 

 meet the surface at the same angle. For our present pur- 

 pose these cases need not be considered ; nor, indeed, do 

 they usually admit of the same kind of treatment as 

 problems in celestial illumination ; simply because bodies 

 which are exposed to light from many different directions 

 are commonly exposed to a good deal of reflected and 

 scattered light from objects of very different degrees of 



reflective power, and few cases arise where the exact con- 

 sideration of the resulting degree of illumination is worth 

 the trouble it would involve. 



Now, let us consider some results of the laws of 

 illumination of opaque bodies under light from a distant 

 source. 



In the first place, it is obvious that a planet l)oing illu- 

 minated proportionately to the inverse scjuare of the dis- 

 tance will appear intrinsically brighter or fainter in a 

 corresponding degree, neglecting for the moment the diffe- 

 rent rert(>ctive powers of different surfaces) from whatever 

 station it may be viewed. Its In-ightness as a whole will, 

 of course, depend on the position of the observer, but its 

 intrinsic brightness will be independent of this considera- 

 tion. Thus, supposing that with the same telescope an 

 observer views in succession all the planets from Mercury 

 to Neptune, and that reducing the field of view he sees 

 only a small part of the planet's disc, on which the smi's 

 rays tixre falling square!//, then supposing the planets to 

 have surfaces of similar reflective power, these successive 

 areas would differ in lustre as — 



1 1 1 



(Mercury's dist)- " (Venus's dist|- ' (Mars' dist)'^ ' &<^-> 



wherever the planet might be placed. Of course the 

 italicised pi-ociso could not be attended to in the case of 

 Mercury and Venus if these planets were in the nearer 

 half of their orbit ; but with these exceptions it is always 

 possible to bring a part of a planet which is under square 

 solar illumination into a small field of view. However, in 

 the preceding remarks I have not had in view the descrip- 

 tion of an experiment to be tried, but rather the illustration 

 of a certain relation. 



It follows that so far as the effects of actual solar illu- 

 mination are concerned — that is, setting aside the reflective 

 qualities of the different planets — we should have the 

 intrinsic brilliancy of the planet's surfaces related in the 

 following manner (where the earth's illumination, when at 

 her mean distance, is taken as unity): — ■ 



In Pcrilielion. Aphelion. 



Mercurv 10-58 ... 4-50 



Venus.'. IW ... I'Jl 



Earth 103i ...- 0'9G7 



Moon 1034 ... 0967 



Mars 0-521 ... 0360 



Jupiter 00408 ... 0-0330 



Saturn 0-0123 ... OOWia 



Uranus 00027 ... 0002.5 



Neptune O'OOll ... OOOll 



It will be observed that the moon is here included 

 as the earth's fellow planet rather than as a satellite. It 

 ought, indeed, to be held in view that the moon really 

 circuits the sun as a planet would do, undergoing rather 

 considerable perturbations, but traversing what may be 

 properly described as an elliptic orbit, having the same 

 aphelion and perihelion as the earth's. 



We shall presently see that the tabic just given, -which 

 ia, of course, the ordinary table of the light received by 

 the several planets, has to be considerably modified to 

 indicate the actual apparent brilliancy (intrinsic) of the 

 planetary surfaces under direct solar illumination. The 

 table is useful, however, as indicating what may be called 

 the normal ratios of planetary intrinsic brightness. 

 (To he continiied.) 



HusoARiAN State Railways. — The net profit realised 

 in 1881 upon the Hungarian State lines was £681,418. 

 This was equivalent to a return upon the capital engaged 

 in the lines at the rate of 2-10 per cent, i er annum 



