TABLE OF CONTENTS. Xlll 



SECTION V. 

 FINITE EXPRESSION OF THE RESULT OP THE SOLUTION. 



ART. PAGE 



205, 206. The temperature at a point of the rectangular slab \vhose co 

 ordinates are x and y, is expressed thus 



SECTION VI. 

 DEVELOPMENT OF AN ARBITRARY FUNCTION IN TRIGONOMETRIC SERIES. 



207 214. The development obtained by determining the values of the un 

 known coefficients in the following equations infinite in number : 



A = 



C = a + 2 5 b + 3 5 c + 5 d + &c. f 

 D = a + 2 b + 3 7 c + 47d + Ac., 

 Ac., &c. 



To solve these equations, we first suppose the number of equations to be 

 m, and that the number of unknowns a, b, c, d, &c. is m only, omitting 

 all the subsequent terms. The unknowns are determined for a certain 

 value of the number ni, and the limits to which the values of the coeffi 

 cients continually approach are sought; these limits are the quantities 

 which it is. required to determine. Expression of the values of a, 6, c, d, 

 &G. when m is infinite ......... 168 



215, 216. The function $(x) developed under the form 



sin2o; + c 



which is first supposed to contain only odd powers of x . . . .179 

 217, 218. Different expression of the same development. Application to the 



function e x - e~ x . . . ..... . . . 181 



219 221. Any function whatever &amp;lt;p(x) may be developed under the form 



^ sin + a 2 sin^x + Og sin3.z+ ... +0^ sin j x + Ac. 

 The value of the general coefficient a&amp;lt; is - / dx &amp;lt;f&amp;gt; (x) sin ix. Whence we 



7T J 



derive the very simple theorem 

 ^ &amp;lt;() = sin a: /&quot;&quot;da 0{a) sina -f sm2xj ^da^a) sin2a + sin3a; /&quot;&quot;da^a) sin3a + &c., 



IT f=3 . r 1 * 



whence 0(x) = S sin ix / da&amp;lt;f&amp;gt;(a.) sin fa .... 184 



2 t=i J o 



222, 223. Application of the theorem : from it is derived the remarkable 

 series, 



- cos x = sin x + sin 4.r + sin 7x + - sin 9^; + &c. . . 188 



*i . A *9 . D.I v 



