J. D. Everett on Reducing Observations of Temperature. 29 
the temperatures of their respective centres, in the constant ratio 
of sin Se ~ each of the portions being supposed to be the 
ith part of a year, where m may be either a whole number or a 
fraction. 
Hence the annual range as shown by the curve of monthly 
mean temperatures will be less than that exhibited by the curve 
of daily mean temperatures in the ratio of sin 15°: are 15°. 
Strictly speaking, instead of ‘(daily mean temperatures,” I 
ought to say “instantaneous temperatures;” but the difference 
is so small as to be quite inappreciable, since the former are to 
the latter nearly in the ratio of sine to arc of 30 minutes or of 1 
to 1:000013. 
' Assuming then that the expression for instantaneous tempera- 
ure is 
y= A-+asin (x+E) 
the mean temperature Y,, of any “th portion of a year will be 
given by the equation 
sin = 
Y,=A+a. sin (e+ E) 
, m 
x being the time for the centre of the portion. Hence if the 
Instantaneous temperatures follow a simple harmonic law, the 
mean temperatures of equal intervals of time will also follow a 
simple harmonic law. For the mean temperature of any peri 
of 80,5, days we have | 
sin 15° . 
Y,2=A+a. mee” Si (+E). 
Secondly, let the expression for instantaneous temperatures be 
y= A,-+a, sin (e+E,)+ <4, sin(2e+E,). 
The expression for the area bounded by two ordinates whose 
distance is 2c will as in the former case fo the integral of ydx 
between the limits a/—c and 2’-+c 
as wee 
= 2A,c+ 2a, sine sin (a’+E,) + 2a, = * sin (22’+- E,) 
and dividing by 2c we obtain for the mean value of y the ex- 
pression ; 
2c 
Hence the mean temperature of any ~th portion of a year is 
Siven by the equation 
Agha, 8S sin (e+ B,) +42 “5 2¢ «in (2 +E,). 
t 
