150 THEORY OF HEAT. [CHAP. III. 



184. Applying the same process to tlie equation 



y = sin #4- - sin 2x 4- ~ sin 5x 4 - sin 7x 4 &c., 

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we find the following series, which has not been noticed, 



-- TT = sin x 4 ^ sin ox 4 - sin ox 4 = sin 7x + -,- sin 9. 4- &c. l 

 4 3 o 7 



It must be observed with respect to all these series, that 

 the equations which are formed by them do not hold except 

 when the variable x is included between certain limits. Thus 

 the function 



cos x -^ cos %x 4 v cos 5x ^ cos 7x + &c. 

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is not equal to JTT, except when the variable x is contained 

 between the limits which we have assigned. It is the same 

 with the series 



sin x - sin 2x 4- sin %x -r sin 4# 4 - sin ox &c. 

 23 4 o 



This infinite series, which is always convergent, has the value 

 \x so long as the arc x is greater than and less than TT. But 

 it is not equal to %x, if the arc exceeds TT; it has on the contrary 

 values very different from \x ; for it is evident that in the in 

 terval from x TT to x = 2ir, the function takes with the contrary 

 sign all the values which it had in the preceding interval from 

 x = to x = TT. This series has been known for a long time, 

 but the analysis which served to discover it did not indicate 

 why the result ceases to hold when the variable exceeds TT. 



The method which we are about to employ must therefore 

 be examined attentively, and the origin of the limitation to which 

 each of the trigonometrical series is subject must be sought. 



185. To arrive at it, it is sufficient to consider that the 

 values expressed by infinite series are not known with exact 

 certainty except in the case where the limits of the sum of the 

 terms which complete them can be assigned ; it must therefore 

 be supposed that we employ only the first terms of these series, 



1 This may be derived by integration from to ir as in Art. 222. [R. L. E.] 



