SECT. VI.] TRIGONOMETRICAL DEVELOPMENTS. 193 



which contains only odd powers of the variable, and we may sup 

 pose it to be developed in the form 



a -j- b cos x -f c cos 2x + d cos Sx -f &c. 



Applying the general equation to this particular case, we find, 

 as the equation required, 



1 . 1 cos 2# cos 4# cos 



_ __.._..__..__ 





 _&&amp;lt;.. 



Thus we arrive at the development of a function which con 

 tains only odd powers in a series of cosines in which only even 

 powers of the variable enter. If we give to a? the particular value 

 JTT, we find 



111111 



5 7r== 2 + rjr375 + o\7- f T9 + 



Now, from the known equation, 



we derive 

 1 



and also 



1111 



^ 7T = 



-&c. 



2 3.5 7.9 11.13 

 Adding these two results we have, as above, 

 111111 1 



T 7T = 7^ + ^ ^ &quot;^ + ~ -^ ^ pr + TT r^ &C. 



4 2 1.3 3.o o.7 7.9 9.11 



226. The foregoing analysis giving the means of developing 

 any function whatever in a series of sines or cosines of multiple 

 arcs, we can easily apply it to the case in which the function to be 

 developed has definite values when the variable is included 

 between certain limits and has real values, or when the variable is 

 included between other limits. We stop to examine this particular 

 case, since it is presented in physical questions which depend on 

 partial differential equations, and was proposed formerly as an ex 

 ample of functions which cannot be developed in sines or cosines 

 F. H. 13 



