249 ON THE LIGHT OF THE MOON AND OF THE PLANET JUPITER. 
or very nearly 
(20) M, = MP. 
According to Lambert's formula, we shall find from (15) and (12) 
_ TRA; Sin. v — v cosu — sin. v — Y COS. v 
qn me y a? T iris. m sin iv : 
Hence we have 
3 sin.? 3 v. ^ Euler and Herschel.- 
Au. 
(22) M = M smi 3 v. Assumed. 
M, = Mi & 
m sin. iv 
These will be compared with the results of Herschel’s observations. In the first 
column of the following statement, the date of the observation is given; in the second, 
the number of stars compared with the Moon; in the third, its elongátion from the 
Sun; in the fourth, the logarithm of the observed moonlight; in the fifth and sixth, the 
altitude and corresponding log. cor. for extinction; next, the logarithm of the observed 
quantity of moonlight corrected for extinction, and referred to the mean full Moon as a 
unit, with the weight given to the determination; in the remaining columns are the 
values of the logarithms of Mj, Mh, etc., calculated according to the above hypotheses. 
M, has been derived from Herschel's log. m*,* by subtracting the constant logarithm 
3.500, and applying a correction for atmospheric extinction. 
log. M, = log. u* — 3.500 + log. cor. for extinction. 
In the correction for extinction, which corresponds to the altitude adopted in the 
adjacent column, it has been assumed that the average altitude of the stars was the 
same as that of the Moon. Its influence is at all events scarcely appreciable, excepting 
in one or two instances. It will be seen from the comparisons which follow, that the 
constant 3.500 applied as above to log. u* is a sufficient approximation for the present 
purpose. The value 3.543 was subsequently found, but its introduction would not 
affect the conclusions arrived at. | 
* Herschel’s p is here designated as p*, 
to distinguish it from the sam 
e lette: i 2 "fos 
cation elsewhere in this Memoir. T uod Wilh a Soren sigil 
