xxii TABLE OF CONTENTS. 



ART. 



411. Integral of equation (e) of vibrating elastic surfaces .... 419 



412. Second form of the integral 421 



413. Use of the same theorem to obtain the integrals, by summing the 

 series which represent them. Application to the equation 



dv d z v 



Integral under finite form containing two arbitrary functions of t . . 422 



414. The expressions change form when we use other limits of the definite 

 integrals 425 



415. 416. Construction which serves to prove the general equation 



417. Any limits a and b may be taken for the integral with respect to a. 

 These limits are those of the values of x which correspond to existing 

 values of the function f(x). Every other value of x gives a nul result 

 forf(x) 429 



418. The same remark applies to the general equation 



the second member of which represents a periodic function . . . 432 



419. The chief character of the theorem expressed by equation (#) consists 

 in this, that the sign / of the function is transferred to another unknown 



a, and that the chief variable x is only under the symbol cosine . . 433 



420. Use of these theorems in the analysis of imaginary quantities . . 435 



421. Application to the equation -^ + ^4 = . . . . . .436 



dx* dy* 



422. General expression of the fluxion of the order t, 



423. Construction which serves to prove the general equation. Consequences 

 relative to the extent of equations of this kind, to the values of / (x) 

 which correspond to the limits of x, to the infinite values of f(x). . 438 



424 427. The method which consists in determining by definite integrals 

 the unknown coefficients of the development of a function of x under 

 the form 



is derived from the elements of algebraic analysis. Example relative to 

 the distribution of heat in a solid sphere. By examining from this 

 point of view the process which serves to determine the coefficients, we 

 solve easily problems which may arise on the employment of all the terms 

 of the second member, on the discontinuity of functions, on singular or 

 infinite values. The equations which are obtained by this method ex 

 press either the variable state, or the initial state of masses of infinite 

 dimensions. The form of the integrals which belong to the theory of 



