XX TABLE OF CONTENTS. 



ART. 



370, 371. Bemarks on different forms of the integral of the equation 



du d?u 



SECTION II. 



OF THE FEEE MOVEMENT OF HEAT IN AN INFINITE SOLID. 



372 376. The expression for the variable movement of heat in an infinite 

 solid mass, according to three dimensions, is derived immediately from 

 that of the linear movement. The integral of the equation 



dv _ d?v d 2 v d 2 v 

 Tt ~ dx* + dy* + &amp;lt;P 



solves the proposed problem. It cannot have a more extended integral j 

 it is derived also from the particular value 



v = e~ n2t cos nx, 

 or from this : 



which both satisfy the equation = ^ . The generality of the in- 



tegrals obtained is founded upon the following proposition, which may be 

 regarded as self-evident. Two functions of the variables x, y, z, t are 

 necessarily identical, if they satisfy the differential equation 



dv d s v d z v d s v 

 dt = dx? + dy* + ~dz? 



and if at the same time they have the same value for a certain value 

 of t ....... &quot; ....... 



377 382. The heat contained in a part of an infinite prism, all the other 

 points of which have nul initial temperature, begins to be distributed 

 throughout the whole mass ; and after a certain interval of time, the 

 state of any part of the solid depends not upon the distribution of the 

 initial heat, but simply upon its quantity. The last result is not due 

 to the increase of the distance included between any point of the mass 

 and the part which has been heated; it is entirely due to the increase 

 of the time elapsed. In all problems submitted to analysis, the expo 

 nents are absolute numbers, and not quantities. We ought not to omit 

 the parts of these exponents which are incomparably smaller than the 

 others, but only those whose absolute values are extremely small . 



383 385. The same remarks apply to the distribution of heat in an infinite 

 solid . * * t ...... .... 



SECTION HI. 



THE HIGHEST TEMPERATURES IN AN INFINITE SOLID. 



386, 387. The heat contained in part of the prism distributes itself through 

 out the whole mass. The temperature at a distant point rises pro 

 gressively, arrives at its greatest value, and then decreases. The time 



