234 ON THE LIGHT OF THE MOON AND OF THE PLANET JUPITER. 
If we extend the integral of sin. d dn over the whole surface of the hemisphere 
upon which s shines, taking 4 for the radius, it becomes 
Tan $dp' = Tô’, 
which gives from (8) 3 
fava +, fingar = pt, 
as in (9). 
The case of most astronomical interest is that of a sphere illuminated by parallel 
rays. Since the diameters of the heavenly bodies are small compared with their dis- 
tances from each other, we may, without sensible error, obtain the amount of sunlight 
incident upon a planet from the formula given above, viz. : — 
(10) CS SC 
where I is the whole light emitted by the Sun, p the semi-diameter of the planet, and r 
its distance from the Sun. : 
If we take de for an element of surface at the Earth presented perpendicularly to 
its radius vector, R, we obtain from (3) for the amount of sunlight incident upon d e, 
(11) dla = ae 
To find the proportion of sunlight reflected back from a planet upon the Earth 
compared with that which the latter receives directly from the Sun, we may substi- 
tute in (5) the value of obtained from (10) ; this will give for the light of the planet 
incident upon de, if the latter is presented perpendicularly to the line 4 joining the 
Earth and planet, | 
va ME Tp de 
(12) deos l6r ri Ai 
Hence, 
(13) dí gr EP 
db A AE 
will represent the ratio of the light reflected from the planet upon the earth, to that 
which the latter receives from the Sun. 
The attempts which have been made to compute the proportion between full moon- 
light and sunlight upon theoretical principles, supposing none to be absorbed by the 
Moon, present singular discrepancies. Thus we have, according to different author- 
ities, ; 
