Horiiiouic AndlysiH. 405 



so tliat 



C, = ^7i .sin(o)/— 6'i)— — ^-^, — ^' -f ^^ i-' — etc. 



3- o- 



But Ci=:(^^ wlieii uit — o therefore O^^o; 



and C\ — /i when w/=7r/2, therefoi-e 



/^ = ,?j[l + 1/3H 1/5' + 1/7' + etc.]=c7,Tr'^iS 



hence c7i = 8///V' and the Fourier series required is 



^ 8//r . _, sinSo)/ sinSco/ ^ -, 

 Cj = ^|sinoj/— --—3 1- — -- —etc. J 



= /^N{oj/') say. 



If the left extremity of the base were at a distance a from the 

 origin instead of coinciding with it, then 



C\ = ;7N(w/— a). 

 Let us call the function /zN(w/ — a) an isosceles function, and the 

 series of isosceles triangles which is the graph of /iN{(Df—a) an 

 isosceles wave specified l)y /i its altitude, and a its phase. 



11. Any wave containing only odd harmonics whose form is 

 polygonal, with n vertices per half wave, can be resolved into 

 fi isosceles waves of the same period, and hence can be ana- 

 lytically represented l)y a sum of /i isosceles functions. 



A vertex may or may not occur where the wave crosses the 

 axis of abscis.sae. In the latter case the base angles of the 

 polygon will be equal. 



For the sake of definiteness let us consider the case when the 

 polygon has 4 vertices per half wave, and let it be specified 

 by Wj, ///.2, m.^ W4, w,5, (///,= — Wj), the tangents of the angles 

 its sides, taken in the positive direction, make with the axis of x^ 

 and by the abscissae x^^, Xjg, x.^^^ x^, of its vertices. 



In the first place let us determine the form of the wave got by 

 adding to the above the isosceles wave /=:^N(w/— a) specified in 

 the above manner by M, — M, and X, so that ^=M7r/2 and 

 a+7r/2^X. In general the new wave will have 5 vertices, 

 the abscissa of the one introduced being X, while the abscissae 

 of the others are unchanged, and if X lie say between :V2a and 

 0:34, and the tangents of the slopes of the sides of the new polygon 

 be «i, «.,, «.^, n^^ «4, — «i, then, remembering that the equations 

 of the different sides are of the form 



