158 THEORY OF HEAT. [CHAP. III. 



of the source of heat; for the source must necessarily supply 

 continually the heat which flows into the masses B and C. If 

 this compensation did not exist at each instant, the system of 

 temperatures would be variable. 



194. Equation (a) gives 

 K - 7 V = (e~ x cos y e~ sx cos 3y + e~ r x cos oy - e~&quot; x cos 7y + &c.); 



CLJC 7T 



multiplying by dy, and integrating from 2/ = 0, we have 



- ( e~ x sin y - e~ 5x sin 3y + - e~ 5x sin oy ^ e~ 7 * sin 7y -f &c. ] . 



If y be made = JTT, and the integral doubled, we obtain 



87T/ 1 _ sv 1 _. x 1 

 \e 4-^e fg + 7 



as the expression for the quantity of heat which, during unit of 

 time, crosses a line parallel to the base, and at a distance x from 

 that base. 



From equation (a) we derive also 

 K -j- = -- (e~ x sin y e~ Bx sin Sy + e~ zx sin oy e~ lx sin 7y + &c.) : 



hence the integral I K I -j- j dx, taken from x = 0, is 

 r {(1 - e~&quot;) sin ?/ - (1 - e&quot; 3: &quot;) sin 3?/ + (1 - e&quot; *) sin 5y 



If this quantity be subtracted from the value which it assumes 

 when x is made infinite, we find 



- ( e~ x sin y - e~ 3x sin Sy + ^ e~* x sin oy &c. ) ; 



7T \ O O / 



and, on making ?/ = j7r, we have an expression for the whole 

 quantity of heat which crosses the infinite edge C, from the 

 point whose distance from the origin is x up to the end of the 

 plate ; namely, 



