xiv TABLE OF CONTENTS. 



ART. PAGE 



224, 225. Second theorem on the development of functions in trigono 

 metrical series : 



-^(o5)=S cosix r n dacosia\!/(a). 



* i=0 Jo 



Applications : from it we derive the remarkable series 

 1 . t 1 cos2x cos 4x 



226 230. The preceding theorems are applicable to discontinuous functions, 

 and solve the problems which are based upon the analysis of Daniel 

 Bernoulli in the problem of vibrating cords. The value of the series, 



sin x versin a + ~ ski 2x versin 2 a + ^ sin 3x versin 3 a -f &c. , 



is ^ , if we attribute to # a quantity greater than and less than a; and 



the value of the series is 0, if x is any quantity included between a and |TT. 

 Application to other remarkable examples ; curved lines or surfaces which 

 coincide in a part of their course, and differ in all the other parts . . 193 



231 233. Any function whatever, F(x), may be developed in the form 



. 

 p) + ^ sina; + Z&amp;gt; 2 sin 2 -f 6 3 sin 3a + &c. 



Each of the coefficients is a definite integral. We have in general 

 2irA = f*&quot;dx F(x) , ira&amp;lt; = f*JdxF(x) cos ix, 



and irb t f dx F(x) sin ix. 



We thus form the general theorem, which is one of the chief elements of 

 our analysis : 



i=^+co / .,J.jj /*.Xf X 



2irF(x) = S (cos ix I daF(a) cos ia + sin ix J daF(a) sin ia ) , 



i= eo \ J TT If J 



i=+oo P + ir 



or 2irF(x) = 2 I daF(a)coa(ix-id) 199 



=_ - 



234. The values of F(x) which correspond to values of x included 

 between - TT and + TT must be regarded as entirely arbitrary. We may 

 also choose any limits whatever for ic ....... 204 



235. Divers remarks on the use of developments in trigonometric series . 206 



SECTION VII. 

 APPLICATION TO THE ACTUAL PEOBLEM. 



236. 237. Expression of the permanent temperature in the infinite rectangular 

 slab, the state of the transverse edge being represented by an arbitrary 

 function .... 209 



