SECT. IV.] EXPENDITURE OF THE SOURCE OF HEAT. 157 



what is the quantity which, during a given time, passes across 

 the base A and replaces that which flows into the cold masses 

 B and (7; we must consider that the flow perpendicular to the 



axis of y is expressed by K^-. The quantity which during 

 the instant dt flows across a part dy of the axis is therefore 



and, as the temperatures are permanent, the amount of the flow, 

 during unit of time, is K-j-dy. This expression must be 



integrated between the limits y = \-rr and y = 4- JTT, in order 

 to ascertain the whole quantity which passes the base, or which 

 is the same thing, must be integrated from y to y = JTT, and 



the result doubled. The quantity -,- is a function of x and y, 



CLJO 



in which x must be made equal to 0, in order that the calculation 

 may refer to the base A, which coincides with the axis of y. The 

 expression for the expenditure of the source of heat is there 



fore 2lfKj-dy}. The integral must be taken from y = Q to 



y = ITT ; if, in the function -j- , x is not supposed equal to 0, 



but x = x, the integral will be a function of x which will denote 

 the quantity of heat which flows in unit of time across a trans 

 verse edge at a distance x from the origin. 



193. If we wish to ascertain the quantity of heat which, 

 during unit of time, passes across a line drawn on the plate 



parallel to the edges B and C, we employ the expression K -j~ , 



j 

 and, multiplying it by the element dx of the line drawn, integrate 



with respect to x between the given boundaries of the line ; thus 



the integral If K -j- dx) shews how much heat flows across the 

 A dy J 



whole length of the line ; and if before or after the integration 

 we make y = \TT, we determine the quantity of heat which, during 

 unit of time, escapes from the plate across the infinite edge C. 

 We may next compare the latter quantity with the expenditure 



