1882.] fluctuations in a variable. 191 



supply. Thus if we call h the heat-supply we have h = ka e (a 1 — 1) ; 



-j- , -j- 2 are positive ; and fluctuations in t about a mean, indicate 



an increased mean value of h ; while conversely, fluctuations in h 

 about a mean, produce a diminished mean of the values of t. Now 

 the materials of the earth's surface have no great capacity for heat. 

 We find that the diurnal and annual variations of heat-supply 

 produce variations of temperature which are of equal period 

 though somewhat retarded in phase. I conclude that the vicissi- 

 tudes of day and night, and of summer and winter, lower the mean 

 temperature of dry land below that which it would have if it 

 received the same amount of heat as now, but uniformly and 

 continuously. 



The capacity for heat of water is considerable ; but it parts 

 with heat communicated to it largely by means of evaporation as 

 well as radiation. Other things being equal the amount of this 

 evaporation at any temperature must bear some relation to the 

 quantity of vapour which Avill saturate the air at that temperature. 

 It is usually considered to be proportional to that. The quantity 

 is a function of the temperature which increases with it under 

 a rapid acceleration. Let h measure the rate of heat supply, f(t) 

 and <£ (t) the rate of heat-emission by radiation and evaporation re- 

 spectively. Then for a body of such small capacity, or otherwise so 

 constituted, that its emission at every moment is equal to its 

 absorption, we have 



h=f(t) + cf>(t), 



where f and </> are functions whose first and second differential co- 

 efficients are both positive. Differentiating we get 



df=f + * 



and therefore positive ; 



§=-</"+*")(/'+*-)-. 



and therefore negative. Hence as before oscillations of temperature 

 about a mean, require a heat-supply whose mean value is greater 

 than that required to maintain the body at that mean temperature ; 

 and conversely, fluctuations of heat-supply about a mean produce 

 a mean temperature which is lower than that which the mean heat- 

 supply would maintain if constant. 



It is not probable that the rays of the sun penetrate very deeply 

 into the sea. And the lightness of watery vapour tends to make 



