192 THEORY OF HEAT. [CHAP. III. 



This and the preceding theorem suit all possible functions, 

 whether their character can be expressed by known methods of 

 analysis, or whether they correspond to curves traced arbitrarily. 



225. If the proposed function whose development is required 

 in cosines of multiple arcs is the variable x itself ; we may write 

 down the equation 



1 



TTX = a + ttj cos x + a 2 cos Zx -f a 3 cos ox+ ... + a t cos ix + &c., 



and we have, to determine any coefficient whatever a it the equa 

 tion a t = I x cos ix dx. This integral has a nul value when i is 



o 



2 

 an even number, and is equal to -^ when i is odd. We have at 



the same time a = 7 ?r 2 . We thus form the following series, 



1 A cos x . cos 3# , cos 5% . cos 7x 



x = ~ TT 4 4 ^ 4 ^3 4 -^ &c. 



2 7T d 7T O7T / 7T 



We may here remark that we have arrived at three different 

 developments for x, namely, 



1 1111 



- x sin x ^ sin 2x + - sin 3# -r sin ^x + - sin 5x &c., 

 tj jb o 



12. 2 2 



- x = - sin oj ^ sin 3^ + r^ sin 5^c - &c. (Art. 181), 



2 TT 3V 5V 



112 2 2 



^X = jTT COSOJ ^ COS &amp;lt;$X -^ COS 5x &C. 



2 4 TT 3V 5V 



It must be remarked that these three values of \x ought not 

 to be considered as equal; with reference to all possible values of 

 x, the three preceding developments have a common value only 

 when the variable x is included between and JTT. The con 

 struction of the values of these three series, and the comparison of 

 the lines whose ordinates are expressed by them, render sensible 

 the alternate coincidence and divergence of values of these 

 functions. 



To give a second example of the development of a function in 

 a series of cosines of multiple arcs, we choose the function sin a?, 



