of Reducing Observations of Temperature. 33 
tance between the ordinates be 2c, then the abscisse of Q and 
R willbe v—candax +c. 
eels Ss 
BE Ss 
The area PQRS is the integral of ydx between the limits 
“z—candx+e 
= a cos {@ —c) — cos (# + e) | = 2asinasine = 2 sine. y 
if 7 be the middle ordinate. 
Hence the area varies directly as 7.—Q. E. D. 
Theorem I1.—If the equation to the annual curve of tem- 
perature be y = a sin zw, the mean temperature of any = th 
part of a year varies directly as the temperature of its middle 
day. The value of m may be either integral or fractional. 
Proof.—Let QR in last figure represent the th part of a 
year, then we have 2c = QR = =a C=. 
m m 
The mean temperature of the period represented by QR is 
the mean height of the figure PQRS ; in other words, is the 
quotient of the area PQRS by the breadth QR. 
But Qsince.y+%=——+y= 
1 
Hence the mean temperature of any cs th of a year is to the 
temperature of its middle day as sin Pees 
mm 
If m = 12, this ratio becomes 1: 1:0115. 
It obviously follows by transformation of co-ordinates, that 
if the equation to the annual curve be 
y = a, + a, sin (w + €,), 
the mean temperature of any =-th part of a year is 
NEW SERIES.—VOL. XIV. NO. 1.—suULY L861. E 
