236 ON THE LIGHT OF THE MOON AND OF THE PLANET JUPITER. 
Referring to the particular cases considered on p. 233, we find that (b) represents the 
proportion of moonlight, if the Moon were a flat, opaque disc, reflecting light upon the 
surrounding concave in proportion to the apparent area of the disc as seen from differ- 
ent points; (g) is the proportion for a disc, reflecting, as in (7), with uniform intensity 
over the whole hemisphere to which it is presented ; while (f£) gives the proportion, 
supposing the Moon's surface polished. : 
Eulers result implies the value © = 4, which is not at all probable. It will be 
noticed that he has supposed that an element, d S, of the surface of the Sun, or of any 
self-luminous body, diffuses its light in equal quantities in all directions. It would 
follow that a sphere self-luminous should appear near its margin, where each element, 
d S, is seen much foreshortened, proportionally brighter than at its centre; but this is 
not in reality true of the Sun,* nor of other bodies shining by their own light. If, on 
the other hand, equal areas are found experimentally to have the same apparent inten- 
sity at all angles of inclination of the surface to the line of sight,T the light from a 
self-luminous sphere like the Sun will be just one half as bright as it would be under 
Eulers hypothesis, if the intensity at perpendicular emission is the same in both 
instances, — being in the one case proportional to the projected area, and in the other, 
to the actual area of the visible hemisphere, or as 
p being the semi-diameter. 
We will now consider the variations in the quantity of light reflected by the Moon 
or a planet produced by changes of phase, and shall find in the course of the inves- 
tigation an explanation of the value 9 — $. An element of its spherical surface pre- 
SR sented at right angles to the parallel incident rays 
> of the Sun, receives more light than an equal 
element would on any other part. The quantity 
received at other angles of incidence will vary as 
SLU the cosine of the angle, and will vanish at the 
margin of the hemisphere. If the Moon is 
viewed from different positions, both the average 
brightness of the illuminated phase, and the rela- 
tive intensity in different parts of it, will change | 
with the angle between the line of sight and of 
d 
as’ I 
* Sir J. Herschel, Outlines of Astron., (395). 
T Ibid., Treatise on Light. Enc. Met. p. 347. Beer, Grundriss des Photometrischen Calcüles, pp. 7, 8. 
