SECT. IV.] PERMANENT STATE OF THE RECTANGLE. 159 



which is evidently equal to half the quantity which in the same 

 time passes beyond the transverse line drawn on the plate at 

 a distance x from the origin. We have already remarked that 

 this result is a necessary consequence of the conditions of the 

 problem ; if it did not hold, the part of the plate which is 

 situated beyond the transverse line and is prolonged to infinity 

 would not receive through its base a quantity of heat equal to 

 that which it loses through its two edges ; it could not therefore 

 preserve its state, which is contrary to hypothesis. 



195. As to the expenditure of the source of heat, it is found 

 by supposing x = in the preceding expression ; hence it assumes 

 an infinite value, the reason for which is evident if it be remarked 

 that, according to hypothesis, every point of the line A has and 

 retains the temperature 1 : parallel lines which are very near 

 to this base have also a temperature very little different from 

 unity: hence, the extremities of all these lines contiguous to 

 the cold masses E and C communicate to them a quantity of 

 heat incomparably greater than if the decrease of temperature 

 were continuous and imperceptible. In the first part of the 

 plate, at the ends near to B or (7, a cataract of heat, or an 

 infinite flow, exists. This result ceases to hold when the distance 

 x becomes appreciable. 



196. The length of the base has been denoted by TT. If we 

 assign to it any value 2^, we must write \ifj instead of y, and 



77&quot; X 1 



multiplying also the values of a? by ~ , we must write JTT -.- 

 instead of x. Denoting by A the constant temperature of the 

 base, we must replace v by -r . These substitutions being made 

 in the equation (a), we have 

 v = ( e&quot;&quot;** cos --. . - e ~u cos 3 4~, + - e~ ~M cos 5 4,7 



7T \ J.L Z.I 3 1 



-^6 cos7^ + &c.J (). 



This equation represents exactly the system of permanent 

 temperature in an infinite rectangular prism, included between 

 two masses of ice B and (7, and a constant source of heat. 



