92 Proceedings of the Royal Society of Edinburgh. [Sess. 
distance, 0 , and a 0 is the constant coefficient of tan 0 in the power series 
in (tan 0 ) which expresses the refraction.* Hence by differentiation 
( 6 ) 
ds 
— sec" 1 z 
1 + 
cot 2 d log a z 
Mod. sin 1' dz 
d ( cos z) 
ds = cosec 2 z( 1-7747)— — . 
d ( cos z) 
dz 
for z< 85° 
for 2 >85’ 
The differential quotients on the right-hand sides are supposed to be 
expressed in seconds of arc per 1' of zenith distance. They are tabulated 
in refraction tables.f 
Substitute the above values in (4) and introduce the notation 
(7) . . F = 10-*» E = F cos z, G = f(j^*A 
\a( cos z)J 
According to section 5 the numerical values of the coefficients of 
E, F, and (cos S sin £)G differ for the twelve curves (one for each month). 
I therefore introduce instead of each coefficient two factors, the one (e,f, g) 
being constant, the other (/u.) being variable, viz. /xg -1 e , /a.f 1 /, n^g. 
Formula (4) then becomes 
(I*) . . . F(z) = /x 2 -1 (eE+/F) - /q“V( C0S S sin $)G. 
(4) Preliminary Calculations. — In the forty-five years 1868*0 to 
1913*0 the temperature was registered at Glasgow Observatory by a 
photo-thermograph, and the curves were measured at points belonging to 
the thermograph hours, i, which are two minutes earlier than the respective 
hours of Greenwich mean time. These measurements were then reduced 
to the readings of the standard thermometer. Means were taken of all the 
figures obtained in the forty-five years at the same hour and in the same 
month, all incomplete days, 2*7 per cent, of the total number, being dis- 
carded. In this way the twelve mean daily temperature curves published 
in Table I were obtained. 
I smooth these curves so as to be better able to interpolate. But the 
corrections do not exceed 0°*05 F., and generally amount to 0°*02 or less. 
Let at t n apparent solar time, E denote the equation of time, X the 
longitude (West), and let both together with the hour-angle be expressed 
in “ hours” (h) of angle. Hence t h = i h —j h , and == 0 h *03 + (E + X) h . I 
interpolate r for i = t -\-j, where t has the values 0 h , l h , 2 h , . . . and thus get 
the temperature curves at full hours of t. 
* Chauvenet, Spherical Astronomy. 
t Ibid. 
