Edinburgh New Philosophical Journal (1861) Edinburgh New Philosophical Journal edinburghnewphil141861 Year: 1861 of Reducing Observations of Temperature. 33 tance between the ordinates be 2c, then the abscissae of Q and R will be x — c and x + c. P T P T S The area PQRS is the integral of ydx between the limits x — c and x + c = a cos | (x — c) — cos (fo + c) i = 2a sin a? sin c = 2 sin c . y if y be the middle ordinate. Hence the area varies directly as y.—Q. E. D. Theorem II.—If the equation to the annual curve of tem- perature be y — a sin x, the mean temperature of any - th part of a ye
Edinburgh New Philosophical Journal (1861) Edinburgh New Philosophical Journal edinburghnewphil141861 Year: 1861 of Reducing Observations of Temperature. 33 tance between the ordinates be 2c, then the abscissae of Q and R will be x — c and x + c. P T P T S The area PQRS is the integral of ydx between the limits x — c and x + c = a cos | (x — c) — cos (fo + c) i = 2a sin a? sin c = 2 sin c . y if y be the middle ordinate. Hence the area varies directly as y.—Q. E. D. Theorem II.—If the equation to the annual curve of tem- perature be y — a sin x, 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 m year, then we have 2c = OR = — . •. c = - • 47 ^ 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. sm But 2 sin c . y -r- 2c = sin c y = m Hence the mean temperature of any — th of a year is to the m temperature of its middle day as sin — : — r J m m If m = 12, this ratio becomes 1: 10115. It obviously follows by transformation of co-ordinates, that if the equation to the annual curve be y = aQ + ax sin (x + ct), the mean temperature of any — th part of a year is NEW SERIES. VOL. XIV. NO. I. JULY 1861. E
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