Xll TABLE OF CONTENTS. 



SECTION II. 



FIBST EXAMPLE OF THE USE OF TRIGONOMETRIC SERIES IN THE 

 THEORY OF HEAT. 



ART. PAGE 



171 178. Investigation of the coefficients in the equation 



l=a cos x +* cos 3x + ecos 5x + d cos 7x + etc. 

 From which we conclude 



or -r=coso:-5cos3a!: + eos5a5- = cos7#-t- etc. 



o O i 



SECTION III. 

 REMARKS ON THESE SERIES. 



179181. To find the value of the series which forms the second member, 

 the number m of terms is supposed to be limited, and the series becomes 

 a function of x and m. This function is developed according to powers of 

 the reciprocal of m, and m is made infinite ...... 



182184. The same process is applied to several other series . . . 



185 188. In the preceding development, which gives the value of the 

 function of x and m, we determine rigorously the limits within which the 

 sum of all the terms is included, starting from a given term , . . 



189. Very simple process for forming the series 



SECTION IV. 



GENERAL SOLUTION. 



190, 191. Analytical expression of the movement of heat in a rectangular 

 slab ; it is decomposed into simple movements ..... 



192 195. Measure of the quantity of heat which crosses an edge or side 

 parallel or perpendicular to the base. This expression of the flow suffices 

 to verify the solution 



196199. Consequences of this solution. The rectangular slab must be 

 considered as forming part of an infinite plane ; the solution expresses 

 the permanent temperatures at all points of this plane . . . . 



200204. It is proved that the problem proposed admits of no other solu 

 tion different from that which we have just stated .... 



