28 J. D. Everett on Reducing Observations of Temperature. 
First let us suppose the equation of the curve (or the expres- 
sion for the temperature in terms of the time) to be 
y = asing, . 
Let 2c denote the length of the interval BD, and let a’ be the 
value of x for its middle point. Then the values of x for points 
B and D will be w/—c and a’+e, and the area ABDC will be 
the integral of ydx taken between these limits, 
= a(cos 2’ — ¢ — cos 2+ c) 
=2e sin 2’, sine 
<=Z2sm¢..9 
if y’ denote the value of y for the middle point of BD. 
Helen the area bounded by two ordinates whose mutual dis- 
tance is given, varies directly as the ordinate drawn midway be- 
tween them. The areas of portions of the curve below the line 
OX must of course be reckoned as negative. 
Dividing the expressions for the area by 2c we obtain 
sine , 
EOF. 
which is therefore the mean value of y for the given interval. 
Let c=", then 27 denoting a year, the given interval 2c will 
be the ‘th part of a year. Hence the mean temperature of any 
“th part of a year is to the temperature of its middle point as 
sin— : =. If the given interval is the /jth part of a year, this 
ratio becomes sin 15°: arc 15° or 1: 1:0115. | | 
These conclusions have been drawn on the i 5 ag that — 
the expression for the temperature is y=asinz. They will still 
be true if the expression be 
y =a sin (x + E) 
for this change only amounts to removing the origin of codrdi- 
nates along the axis of x and does not alter the values of the 
ordinates. 
If the expression for the temperature be 
y= A-+a.sin(x-+ E) 
the ordinates will be greater than before by the constant quan- 
tity A, which represents the mean temperature of the year; 
hence the temperatures will require to be diminished by the 
mean of the year in order that the above conclusions may be ~ 
applicable. The following theorem will hold in all three cases, 
WZi— 
The difference between the mean temperatures of any two | 
equal portions of the year will be less than the difference between 
