SECT. IV.] UNIFORM LINEAR MOVEMENT. 47 



joins the extremities a. and /3; thus, denoting by z the height of 

 an intermediate section or its perpendicular distance from the 

 plane A, by e the whole height or distance AB, and by v the 

 temperature of the section whose height is z, we must have the 



b a 

 equation v = a -\ -- z. 



6 



In fact, if the temperatures were at first established in accord 

 ance with this law, and if the extreme surfaces A and B were 

 always kept at the temperatures a and b, no change would 

 happen in the state of the solid. To convince ourselves of this, 

 it will be sufficient to compare the quantity of heat which would 

 traverse an intermediate section A with that which, during the 

 same time, would traverse another section B . 



Bearing in mind that the final state of the solid is formed 

 and continues, w r e see that the part of the mass w r hich is below 

 the plane A must communicate heat to the part which is above 

 that plane, since this second part is cooler than the first. 



Imagine two points of the solid, m and m, very near to each 

 other, and placed in any manner whatever, the one m below the 

 plane A , and the other m above this plane, to be exerting their 

 action during an infinitely small instant : m the hottest point 

 will communicate to m a certain quantity of heat which will 

 cross the plane A . Let x, y, z be the rectangular coordinates 

 of the point m, and x, y , z the coordinates of the point m : 

 consider also two other points n and n very near to each other, 

 and situated with respect to the plane B , in the same manner 

 in which m and m are placed with respect to the plane A : that 

 is to say, denoting by f the perpendicular distance of the two 

 sections A and J5 7 , the coordinates of the point n will be x, y, z + f 

 and those of the point n , x, y , z f + % ; the two distances mm 

 and nri will be equal : further, the difference of the temperature 

 v of the point m above the temperature v of the point m will 

 be the same as the difference of temperature of the two points 

 n and n . In fact the former difference will be determined by 

 substituting first z and then / in the general equation 



b a 



and subtracting the second equation from the first, whence the 



