190 Mr Hill, On the effect of [Feb. 20, 



mean value of f{y) for a continuous change in y exceeds or falls 



short of the value for the mean magnitude of y according as -' t is 



positive or negative. 



The same will be equally true if y change back again from 

 PL to QM. Therefore it is true for oscillations of y. Thus, — 

 If a variable perform oscillations about a mean magnitude, the 

 mean value of a function of that variable is greater or less than 

 the value for that mean magnitude, according as the second 

 differential co-efficient of the function is positive or negative. 



Conversely, if u =f(y), 



d 2 y _ d 2 u fdu 

 du* ~ dy 2 \dy 



_££ (it 



dy 2 \dy 



So that if -j- be positive, ~{ is opposite in sign to -7-5 or ~ 2 , and 



therefore if we cause/ instead of y to oscillate about a mean, the 

 resulting mean of y is less than its value for the mean magnitude 

 of/, when/ is such a function that for oscillations of y it is greater 

 than its value for the mean magnitude of y. And vice versa. 

 (This does not hold if / have a maximum or minimum value 

 between the limits of variation.) 



An important instance of this is the connection between 



Radiation and Temperature. According to Dulong and Petit the 



radiation of a body at temperature t + 6 surrounded by an envelope 



of temperature 6 is proportional to the function a e (a ( — 1). Here 



dv ctv 

 if we denote the rate of radiation by r, -j and -rj are both positive. 



at ctz 



Therefore if the temperature of a body be made to oscillate, its 

 mean radiation is greater than its radiation at the mean tem- 

 perature. Conversely if its rate of radiation after being constant 

 under a constant temperature be made to fluctuate about that rate 

 as a mean, the mean temperature must fall below its previous 

 constant value. 



Again, if we have a body receiving heat from some external 

 source, its temperature rises till its radiation equals its absorption. 

 If its capacity for heat be very small, this rise will take place 

 so rapidly as to be nearly simultaneous with the increase in heat- 

 supply, and the radiation at any moment will nearly equal the 



