TABLE OF CONTENTS. XIX 



ABT. PAGE 



The function P (x) satisfies the condition F (x) = F ( - x). Expression of 



the variable temperatures .......... 333 



348. Application to the &amp;lt;case in which all the points of the part heated 

 have received the same initial temperature. The integral 



I sin 2 cos qx is i 

 Jo 







if we give to x a value included between 1 and - 1. 



The definite integral has a nul value, if a; is not included between 

 1 and - 1 ............. 338 



3-49. Application to the case in which the heating given results from the 



final state which the action of a source of heat determines . . . 339 

 350. Discontinuous values of the function expressed by the integral 



34 



351 353. We consider the linear movement of heat in a line whose initial 

 temperatures are represented by vf(x) at the distance x to the right 

 of the origin, and by v = -f(x) at the distance x to the left of the origin. 

 Expression of the variable temperature at any point. The solution 

 derived from the analysis which expresses the movement of heat in an 

 infinite line ..... ...... . ib. 



354. Expression of the variable temperatures when the initial state of the 



part heated is expressed by an entirely arbitrary function . . . 343 



355 358. The developments of functions in sines or cosines of multiple arcs 



are transformed into definite integrals ....... 345 



359. The following theorem is proved : 



!Lf(x} I dqsinqx I daf (a) sinqa. 

 * Jo Jo 



The function / (x) satisfies the condition : 



348 



360 362. Use of the preceding results. Proof of the theorem expressed 

 by the general equation : 



f+ r* 



7T0 (x) = I da &amp;lt;p (a) I dq COS (qx - qa). 

 ./- Jo 



This equation is evidently included in equation (II) stated in Art. 234. 

 (See Art. 397) ib. 



363. The foregoing solution shews also the variable movement of heat in an 

 infinite line, one point of which is submitted to a constant temperature . 352 



364. The game problem may also be solved by means of another form of the 

 integral. Formation of this integral 354 



365. 366. Application of the solution to an infinite prism, whose initial 

 temperatures are nul. Remarkable consequences 356 



367 369. The same integral applies to the problem of the diffusion of heat. 

 The solution which we derive from it agrees with that which has been 

 stated in Articles 347, 348 .... .... 362 



