Harmonic Analysis. 407 



and the complete wave will be fully lepresented by the sum of the 

 n isosceles functions given by 



^j- = n+l r ^ 



- 2 I (w r — W,.+i)N((o/ - .r,., ,+1 + 7r/2), j- 



reuienibering that ///„^i= - ;//j. 



The following are examples of the preceding method. 



12(a). Wave form a trapezium with equal base angles. This 

 is the sum of two equal isosceles waves. 



Take the left extremity of the base of the trapezium as the 

 the origin of abscissae and let it be specified by 



Wj = m, f/i^ = 0, m.^ = — w, rtjo = /x, a.vi = tt — /x. 

 so that its altitude t=.^m. 



By § 11 the expression for the wave is 



^[N(a,/-/x + 7r/2) + N(a)/ + Ai-7r/2)] 



2wr . , , , ,j,, sin3(to/— w. + 7r/2) 

 = — [sin(a>^ - /x + 7r/2)- ^---^ ^Ll + etc. 



, • / . , /o\ sin3((D/ + a — 7r/2) , . T 

 + sm(w/ + /A — 7r/2)— ^ ^^^^ '-'' + etc.] 



itr . ■ ^ , sinSusinStii/ 

 = — I sni/xsinoj/+ '-— 



/ATT 3 



, sin5asin5w/ , , , 

 + ^, +«tc.J, 



which is Fourier's expansion for a wave of this form. 



12(^). Wave foi'm a triangle. This is the difference of two 

 isosceles waves when the vertex of one lies on a side of the other. 



Take the left extremity of the base of the triangle as the 

 origin of abscissae and let it be specified by 



Wj = ;//, Wo^: — ?/, m.j, = — w, c?i2 = /u., a^^ = w 

 so that its altitude /^=:/x;;/^(7r — /a);/, and ;//, ;;, are the tangents 

 of its base angles. 



By Jj 1 1 its expansion in i-sosceles functions is 



^[(w + ;/)N(a>/-/x + 7r/2) + (//^-«)N(co/-7r/2)] 



m( — /i, + 7r/2) - 

 ■N(W-7r/2)] 



= !^'[N(co/-/. + 7r/2)+N(a,/-7r/2)] + -^''[N(a./-/. + 7r/2) 



