TABLE OF CONTENTS. 



ART. PAGE 



after which this maximum occurs, is a function of the distance x. 

 Expression of this function for a prism whose heated points have re 

 ceived the same initial temperature 385 



388391. Solution of a problem analogous to the foregoing. Different 



results of the solution 387 



392 395. The movement of heat in an infinite solid is considered ; and 

 the highest temperatures, at parts very distant from the part originally 

 heated, are determined 392 



SECTION IV. 

 COMPARISON OF THE INTEGRALS. 



396. First integral (a) of the equation -=- = -=- (a). This integral expresses 



the movement of heat in a ring ...... . . 396 



397. Second integral (/3) of the same equation (a). It expresses the linear 

 movement of heat in an infinite solid ....... 398 



398. Two other forms (7) and (5) of the integral, which are derived, like the 

 preceding form, from the integral (a) ....... t 6. 



399. 400. First development of the value of v according to increasing powers 

 of the time t. Second development according to the powers of v. The 

 first must contain a single arbitrary function of t ..... 399 



401. Notation appropriate to the representation of these developments. The 

 analysis which is derived from it dispenses with effecting the develop 

 ment in series ............ 402 



402. Application to the equations : 



d-v d*v d 2 v . d z v d*v , 



^ = d* + d?-- : &quot;- (e)l nd d? + ^= ...... (d) - 404 



403. Application to the equations : 



(/) 405 



404. Use of the theorem E of Article 361, to form the integral of equation (/) 



of the preceding Article .......... 407 



405. Use of the same theorem to form the integral of equation (d) which 

 belongs to elastic plates ......... k 409 



406. Second form of the same integral ........ 412 



407. Lemmas which serve to effect these transformations .... 413 



408. The theorem expressed by equation (E), Art. 361, applies to any number 



of variables ......... ... 415 



409. Use of this proposition to form the integral of equation (c) of Art. 402 . 416 



410. Application of the same theorem to the equation 



d 2 v d-v d-v 



+ + = ...... 41S 



