SECT. VII.] MOVEMENT IN THREE DIMENSIONS. 75 



and then substituting the co-ordinates of M and J/ , we find also 

 V V = a (x x) + b (y y) +c(z /). Now the quantity of 

 heat which m sends to m depends on the distance mm, which 

 separates these molecules, and it is proportional to the difference 

 v v of their temperatures. This quantity of heat transferred 

 may be represented by 



q(v-v )dt; 



the value of the coefficient q depends in some manner on the 

 distance mm, and on the nature of the substance of which the 

 solid is formed, dt is the duration of the instant. The quantity 

 of heat transferred from M to M t or the action of M on M is 

 expressed likewise by q (VV) dt, and the coefficient q is the 

 same as in the expression q (v v) dt, since the distance MM is 

 equal to mm and the two actions are effected in the same solid : 

 furthermore V V is equal to v v, hence the two actions are 

 equal. 



If we choose two other points n and ri, very near to each 

 other, which transfer heat across the first horizontal plane, we 

 shall find in the same manner that their action is equal to that 

 of two homologous points N and N which communicate heat 

 across the second horizontal plane. We conclude then that the 

 whole quantity of heat which crosses the first plane is equal to 

 that which crosses the second plane during the same instant. 

 We should derive the same result from the comparison of two 

 planes parallel to the plane of x and z, or from the comparison 

 of two other planes parallel to the plane of y and z. Hence 

 any part whatever of the solid enclosed between six planes at 

 right angles, receives through each of its faces as much heat as 

 it loses through the opposite face ; hence no portion of the solid 

 can change temperature. 



94). From this we see that, across one of the planes in 

 question, a quantity of heat flows which is the same at all in 

 stants, and which is also the same for all other parallel sections. 



In order to determine the value of this constant flow we 

 shall compare it with the quantity of heat which flows uniformly 

 in the most simple case, which has been already discussed. The 

 case is that of an infinite solid enclosed between two infinite 



