of Reducing Observations of Temperature. 81 
where x is the time for the centre of the part; and let the 
corresponding expression for the “th part of a year be— 
y =a, +a, sin (x+e,)+a, sin (2Qa+e)+ ....+a*sin (ne + ¢,) 
then the following relations will exist :— 
Een Oa, Wy 10, Et, = Oa9 «rt, @ > Ln ae ene 
Le © Rice Di 3 
a2 2, >: Msin yi msn. A,ia,::Msin >-:msin—. 
_ ne . ne 
Ae. 2 Ne Sie" a SI 
M m 
If M=2, the coefficients A, A, A,, &c. vanish (since sin x 
=sin2r=sin 3r7= ....=0). Hence the mean temperature 
of a half year is independent of the terms which involve these 
coefficients. | 
If M is infinite, we have A, : a,::ne:msin —*, which is the 
relation between instantaneous and mean temperatures. 
Demonstration of Theorems stated in Paper on Reduction 
of Observations of Temperature. By Professor J. D. 
EVERETT. | 
Definition.—A simple harmonic curve is one which is 
capable of being expressed by the equation y = a. sin a, 
where a is a constant. The general equation of such a curve 
referred to any origin, but without changing the directions of 
the axes, 1s 
y = 4 + a, sin (x + @,). 
The form of the curve will be as here represented, and the 
curve will extend indefinitely in both directions, continually 
repeating itself. The portion KQLSM of the curve con- 
tains an entire period, during which the quantity w + e, goes 
