SECT. I.] MOVEMENT ACROSS INFINITE PLANES. 361 



hence 



1st, if we suppose x nothing, we find v = 1 ; 2nd, if x not 

 &quot;being nothing, we suppose t = 0, the sum of the terms which 



contain x represents the integral \dre~** taken from r = to r = oo , 



- 



and consequently is equal to \J-jr; therefore v is nothing; 3rd, 



different points of the solid situated at different depths cc lt x v # 3 , 

 &c. arrive at the same temperature after different times t lt t it t & , 

 &c. which are proportional to the squares of the lengths x lt a? 2 , x z , 

 &c.; 4th, in order to compare the quantities of heat which during 

 an infinitely small instant cross a section S situated in the interior 

 of the solid &quot;at a distance x from the heated plane, we must take 



the value of the quantity KS r and we have 



thus the expression of the quantity -T- is entirely disengaged from 

 the integral sign. The preceding value at the surface of the 



/ /Hf/} T7&quot; 



heated solid becomes S _ - , which shews how the flow of heat 



at the surface varies with the quantities C, D, K, t ; to find how 

 much heat the source communicates to the solid during the lapse 

 of the time t, we must take the integral 



