ON THE LIGHT OF THE MOON AND OF THE PLANET JUPITER. 237 
illumination. This angle, which is the external angle at s of the triangle s Ids’, 
Fig. 1, we will call v. | 
From the condition that the amount of sunlight incident upon any element of the 
illuminated hemisphere. of the Moon is proportional to the cosine of the angle between 
the Sun and the zenith of that element, Lambert has obtained the expression * 
2 y sin? l sin? y 
(14) E 
ane oe (sin. v — v cos. v) 
for the ratio of the light of the Moon, at any given phase, to that of the Sun. 
Z is the semi-diameter of the Moon seen from the element d e on the Earth, 
c [11 [11 [11 Sun Di [11 113 113 Moon, 
P « “ 113 Sun DI “ 193 é Earth. 
If we substitute in (14), 
` ^ Sun's semi-diameter : un's semi-diameter 
sin. 15, sin. o = SM sin. $ = S 
A T R 
, 
it becomes 
2u D Rp 
— (SM. Vv — Y COS. v . 
at ) Ar 
Comparing this with (13), we have 
(15) KE an y EE dü _ 2p sin. v — v cos. v Rp 
AS De o x cu 
as the expression, according to Lambert, of © and Ss 
for a planet partially illuminated. When v = 180°, O = 5, as on p. 235. 
Euler,j on the other hand, gives a formula, which, reduced to the same notation, 
becomes 
for the phases of the Moon or 
dí  psn?l, Rp 
(16) — MAL ef d ; 
dL 8 A??? 
hence, 
sek usb 
10) T. sin, 9 v 
The quantity of light received is here assumed to be proportional to the area of the 
illuminated phase, and the average intensity of its bright surface to be the same at 
all ages of the Moon, which is a condition not at all likely to hold good. 
Lamberts solution is the only one] which attempts to represent the gradation in the 
* Photometria. Beer, Grund. des Phot. Cal., p. 69. T Mem. Ac. Ber., 1750. 
i Wollaston, Ph. Tr., 1829, has adopted the same principle, but he has deduced from it an erroneous for- 
mula for the full Moon, viz.: 4 313-3395 oi 
aL: 3 
VOL. VIII. 31 
