Mean Temperature of Rivers above that of the Atmosphere. 357 



represents tbe number of degrees of temperature generated in a 

 mass of water by descending during unity of time along the 

 channel of the river. 



The temperatui'e of the river will rise until the loss of heat by 

 conduction, radiation and evaporation exactly balances the pro- 

 duction of heat by friction. This loss of heat must be approxi- 

 mately proportional to the excess of the temperature of the water 

 above that of the atmosphere. 



Let C represent the loss of heat, in degrees, for one degree of 

 excess of temperature, sustained by unity of weight of water 

 through unity of surface exposed to che air ; 



C the corresponding coefficient for the surface in contact with 

 the bed of the channel. 



Let ]M denote the volume of unity of weight of water, that is 

 to say, 0'016 cubic foot per lb. avoirdupois. 



Let s be the ai-ea of the transverse section of the river ; 

 b the breadth of its surface ; 

 p the peripheiy of its bed. 



Then 



Mb Mp 

 s ' s 



are the areas exposed by unity of weight of water in the channel 

 to the air and to the soil respectively ; and, if 



AT be the excess of the temperature of the river above that of 



the atmosphere, 

 AT' its excess above that of the soil, 



the loss of heat by conduction, radiation and evaporation, in unity 

 of time measured in degrees, will be represented by 



— (CiAT + C/^AT'). 



This quantity being made equal to the gain of heat by friction, 

 we have for the condition of equilibrium of temperature the fol- 

 lowing equation : — 



^=:— (CAAT + C';jAT'). ..... '(1) 



K. s 



If the temperatui-e of the air and of the soil be the same, so 

 that AT = AT', then this equ<ation becomes 



|- = y(C6-hCy)AT, 



AT-— -"^^ - 

 ^^~KM(C6 + C'y) 



(2) 



