SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. Ill 



prisrn will be expressed by the equation 



, dv dv dv 



w = v +-T-+-7-^+ JT~?I 



dx * dy dz 



Thus the molecules infinitely near to the point m will have, 

 during the instant dt, the same actual temperature in the solid 

 whose state is variable, and in the prism whose state is constant. 

 Hence the flow which exists at the point m, during the instant dt, 

 across the infinitely small circle &&amp;gt;, is the same in either solid ; it 



is therefore expressed by K -7 codt. 



CL2 



From this we derive the following proposition 



If in a solid whose internal temperatures vary with the time, by 

 virtue of the action of the molecules, we trace any straight line what 

 ever, and erect (see fig. o), at the different points of this line, the 

 ordinates pm of a plane curve equal to the temperatures of these 

 points taken at the same moment; the flow of heat, at each point p 

 of the straight line, will be proportional to the tangent of the angle 

 a. which the element of the curve makes with the parallel to the 

 alscissw ; that is to say, if at the point p we place the centre of an 



Fig. 5. 



infinitely small circle o&amp;gt; perpendicular to the line, the quantity of 

 heat which has flowed during the instant dt, across this circle, in 

 the direction in which the abscissae op increase, will be measured 

 by the product of four factors, which are, the tangent of the angle 

 a, a constant coefficient K, the area o&amp;gt; of the circle, and the dura 

 tion dt of the instant. 



141. COROLLARY. If we represent by e the abscissa of this 

 curve or the distance of a point p of the straight line from a 



