SECT. VI.] 



GEOMETRICAL ILLUSTRATION. 



201 



i i 



7T(f) (x) = ^ &amp;lt;j&amp;gt; (x] dx + cos x 



ft 



sign I an integral which begins when the variable is equal to a, 



J a 



and is completed when the variable is equal to b ; and we write 

 equation (n) under the following form : 



r* 



(x) cos x dx -f cos 2x (f&amp;gt; (x} cos 2xdx 



Jo 



[n 



+ cos 3x $ (x) cos %xdx + etc ........... (V). 



J 



Instead of taking the integrals from x = to x TT, we might 

 take them from x = to x = 2?r, or from x IT to x = TT; but in 

 each of these two cases, TT&amp;lt;/&amp;gt; (x} must be written instead of JTT^ (a:) 

 in the first member of the equation. 



232. In the equation which gives the development of any 

 function whatever in sines of multiple arcs, the series changes 

 sign and retains the same absolute value when the variable x 

 becomes negative; it retains its value and its sign when the 

 ariable is increased or diminished by any multiple whatever of / 



Fig. 10. 



v 



the circumference 2?r. The are ^a (see fig. 10), which cor 

 responds to the interval from to TT is arbitrary; all the other 

 parts of the line are determinate. The arc &amp;lt;/&amp;gt;(a, which corresponds 

 to the interval from to TT, has the same form as the given arc 

 (fxfra ; but it is in the opposite position. The whole arc OLffxjxfxjxi is 

 repeated in the interval from TT to 3?r, and in all similar intervals. 

 We write this equation as follows : 



- TT&amp;lt; (a;) = sin x I (f&amp;gt; (x) sin xdx + sin 2x I &amp;lt;f&amp;gt; (x) sin Zxdx 

 2 Jo Jo 



+ sin 3x I (j&amp;gt; (x) sin 3xdx + &c. 



