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of Reducing Observations of Temperature. 21 
The curve which is adopted as the standard of reduction is 
what mathematicians call a “simple harmonic curve,” or “the 
curve of sines,” and is expressed by the equation* 
y=A+ta. sin (+e), 
where A will denote the mean temperature of the year, a the 
amplitude or greatest departure of the curve from the mean, 
which will be the same above as below, and will therefore be 
equal to half the annual range, and e is expressive of the 
“date of phase,” the phases of temperature being earlier or 
later according as e is greater or less. The curve has one 
maximum and one minimum in the year, which are precisely 
half a year asunder; and exactly midway between these, are 
the two points where the curve intersects the line of mean 
annual temperature, corresponding to those-two days in the 
year, one in spring and the other in autumn, whose mean tem- 
peratures are the same as that of the year. 
The curve for a year will consist, in fact, of four precisely 
similar portions, the part which is above the line of mean 
temperature being precisely similar to that which is below, 
and each of these halves being divided symmetrically, at 
the points of maximum and minimum temperature respec- 
tively. 
It is not, of course, pretended that the actual temperature 
of any place fulfils these conditions; but that when a uniform 
standard of reduction is to be applied to a number of places 
(in the temperate or frigid zones), such a curve as we have 
described is adapted to the purpose. While possessing the 
necessary amount of uniformity, the curve, at the same time, 
admits of infinite variety in respect of its amplitude (é.¢., the 
extent of its departure from a straight line), which may be 
increased or diminished, without limit, according as we wish 
to represent a climate where the annual range is great or 
small. 
It is not necessary that the curve should actually be drawn. 
* Some of our most important results will remain true if the temperature 
through the year be supposed to conform to the expression 
y=A-+a,.sin (w+e,)+a,. sin (2a+¢,). 
We shall indicate these in their place. 
