96 Proceedings of the Royal Society of Edinburgh. [Sess. 
of b between zero and O’ 10 might be employed. Though zero value of b 
gives the smallest residuals, I arbitrarily adopt 005, which is a better value 
for the night curves than zero, and will be used for them (§ 9). The 
residuals for the value of 0*05 are only slightly larger than for zero value. 
Therefore the values of the constants become 
and herewith 
a-0-538, 6 = 0*050, e = 0-706, /= 0-196, ^ = 0-346, 
a = 0T495 s log e~/ 3t =---/3 Mod. t= - 0 0519 t 
log e-v 4 = -y Mod. t= -0-1818/. 
(8) The Values of the Constant c 0 and of the Constants of Integration . 
— I calculate the values of s from (5) for each full hour of t at which 
the sun stands above the horizon and for each month. I take the average 
temperature and pressure of the air into account (by means of factor A in 
Table II, which gives the principal results for June), though this is perhaps 
an unnecessary complication. With these values of s I calculate further 
E, F, G from (6) and (7), which I substitute in (I*), and find ( 2 a) _1 F( 0 > 
from ( 2 a)- 1 F( 0 ) = (O-3731V- 1 i(9-8166-lO) M - 1 F-(O-O634) / a- 1 cosd sin t G, 
the figures enclosed in brackets being logarithms. 
The functions and X 2 are then found, according to formulas (II), by 
adding 0-0519 t and 0*1818 & The integration is effected by mechanical 
quadrature, the lower limit t r being the full hour nearest sunrise. The 
numerical values of T being thus known (cf Tables II and III), the 
equations (II) furnish for each month from seven (for December) to nine- 
teen (for June) linear equations for the determination of the unknowns 
c 0 , c v c 2 . The solution is best made by trial and error, c 0 being suitably 
chosen and the other two unknowns being calculated. 
(9) The Night Curves . — The differential equation is again given by (I) 
if F(z) be replaced by zero. Its integral is formula (II), where T = 0. 
Also in this case the constants a and b are determined from the linear 
equation (I), in which t, t , r are numerically known. The calculation is 
simple. c 0 is eliminated from the equations of the same month by deduct- 
ing from each equation the mean of all the equations. Then take the sum 
of all the equations belonging to the same month. Combine these in two 
groups, which are : 
April to September . a = 0'47 + 6*7 b 
October to March . a = 0-19 + 7 *3 b 
I adopt .... a = 0-32 + 7-06 
as compared with . . a = 0 - 32+4*4. b for the day curves (§ 6). 
The value of b is again not accurately determinable. If a be substituted in 
