616 
THE EAEL OF EOSSE ON THE E ADI ATI ON 
Comparison of the reducing curve in zenith-distance with. Laplace’s law. 
As the observations of the moon's radiant heat were all made within a range of little 
more than 50° of zenith-distance, it is of interest to see if any of the more generally 
received formulae for the extinction of heat or light in our atmosphere agree with the 
Table at p. 598; for only in the event of this being the case can we extend our consider- 
ations to some interesting cases not included within that range. 
The best form for such a formula is doubtless that first given by Laplace *, and 
which is to be found in Pouillet’s ‘ Meteorologie’ (t. ii. p. 711) in the modified form 
••■(!) 
where t is the heat-effect in a given interval of time, and z the amount of air through 
which the rays of heat have passed ; a and p are constants to be determined from the 
observations ; z is calculated on the assumption that the atmosphere forms a coat of 
uniform density of a thickness h, which is taken equal to unity. If any value, r, for the 
radius of the earth be assumed, then s can be calculated for any zenith-distance, z, by 
the formula 
s = \/ 2 rli -f- Id -j- r 2 cos 2 z — r cos z. 
Substituting logarithms in (1), we have 
log t = log a-j- z log p, 
where for log t we can take <pz 'with its sign changed, and thus get a series of equations : — 
— <pz = log«-(-£ logy*,' 
-(<f>z)' = log a + s' log^, 
y ( 2 ) 
— [<pz)"= log a-V-z" logjp, 
&c. &c. 
whence log a and log p can be deduced. In his researches on the sun’s radiant heat 
Pouillet assumed r— 80 h. In accordance with this the values of s for every degree 
from 29° to 80° were calculated and the constants a and p deduced. It was then found 
that the values of <pz were tolerably well represented, yet at the same time it was evident 
that r=70 h or r=6Q h would fulfil the required conditions more accurately; and in 
point of fact r=60 li is very nearly the value that makes the outstanding errors in the 
representation of (pz a minimum; r=5Sh lessens the sum of the squares of the errors 
to an extent altogether immaterial. The following are the 51 equations of the form (2), 
where the coefficient of log p is 
s=< v /l21 + 3600 cos 2 z — 60 cosz. 
* Mecanique Celeste, t. iv. livrc s. chap. 3. 
