ON THE LIGHT OF THE MOON AND OF THE PLANET JUPITER. 241 
the angles of elongation comprising the observations, a new expression nearly propor- 
tional to sin. 1 v, there will be no difficulty in satisfying the data. It should be 
noticed that this expression is not proposed as a substitute for Herschel's hypothesis, 
but merely to show that the observations in question can be satisfied without resorting 
to the latter. It will subsequently appear that neither of the formule represents 
the entire range of phases from new to full Moon. 
The formula employed by Herschel, to find the proportion between the light from a 
given phase and that of the mean full Moon, may be derived from (12) by the follow- 
ing substitutions : — | 
JM, — the light from the phase compared with the mean full Moon. 
d i, = the quantity of light from the mean full Moon, incident at right angles upon an element de 
at the Earth. 
(17) 7) = the corresponding distance of the Moon from the Sun. 
4, = its mean distance from the Earth. 
|,  — its mean angular semi-diameter. 
o = jeh: Jo. 
R- = Earth’s radius vector. 
(12) then becomes, making at the full phase v, = 180°, 
, Ir g% Malta 
q = TE de, , di wd Ze SC sin? 1 v de, 
and we shall have 
* 242 
(18) M = oy = e sin? $v, 
in which may be used 
ry — 1.0025 i.d 
r — R (1 — 0.0025 cos. v) E 
l being the augmented semi-diameter as seen from de. The difference between the 
Moon's semi-diameter seen from de and from the centre of the Earth, as well as the 
difference between the distance of the Earth and of the Moon from the Sun, are, how- 
ever, too small to have an appreciable influence upon the experiments. 
By the same substitutions, but making 
0 = 3 sin? 4 v, 
we have 
(19) M, = M, sin.‘ 4 v. 
The value of M best representing the observations between quadratures and opposi- 
tion, when the light of the stars is taken as the standard, is, according to Herschel 
M, = M^, 
