SECT. VI.] GEOMETRICAL DEMONSTRATION. 203 



ordinate of or x F . We must draw also the line -vJ/^ ChJ^ whose 

 ordinate measures the half-difference between the ordinate of 

 F F mFF and that of f f mff. This done the ordinate of the 

 lines FF mFF, and f f mff being denoted by F (x) and f(x) 

 respectively, we evidently have /(a?) = F( x) ; denoting also the 

 ordinate of $ $m$$ by &amp;lt; (x), and that of iJrSJr Oi/nJr by ^ (x), 

 we have 



F(x) = &amp;lt;j, (x) + f (x) and f(x) = $(x}-^(x}=F (- x), 

 hence 



&amp;lt; (x) = i* + lF(- x) and + (*) = * - ^(-*), 



whence we conclude that 



&amp;lt;$&amp;gt;(x) = $(-x) and ^ (x) = - ^ (- a?), 

 which the construction makes otherwise evident. 



Thus the two functions (/&amp;gt; (x) and i|r (x), whose sum is equal to 

 F (at) may be developed, one in cosines of multiple arcs, and the 

 other in sines. 



If to the first function we apply equation (v), and to the second 

 the equation (/x), taking the integrals in each case from x = - TT 

 to X = TT, and adding the two results, we have 



2 /(*) ^ + cos x |^{*) cos ^ ^ + cos 2a? /&amp;lt;/) (a;) cos 2% dx + &c. 



+ sin x^r(x} sin re dx + sin 2# ^(#) sin 2aj Ja; + &c. 



The integrals must be taken from x = TT to x = IT. It may now 



f +7r 

 be remarked, that in the integral I &amp;lt; (x) cos a? cfo we could, 



J -IT 



without changing its value, write (x) + -^ (a?) instead of &amp;lt;&amp;gt; (a?) : 

 for the function cos a? being composed, to right and left of the 



axis of x t of two similar parts, and the function ^r (x) being, on the 



r+Tr 

 contrary, formed of two opposite parts, the integral I ty(x) cos xdx 



J -IT 



vanishes. The same would be the case if we wrote cos 2a; or 

 cos 3a-, and in general cos ix instead of cos a?, i being any integer 



