Harmonic Analysis. 397 



Joi yii J'li yn-i be n equi-spaced ordinates that exactly in- 

 clude half the wave, i.e., ordinates corresponding to the al)scissae 

 X, x + ir/fi, a* + 2 7r/«, .... x + fi-lirjft respectively, and 



called e.s. ordinates in the sequel ; and if N^, Hj, N.^ N„.i, 



be the ordinates of the ;/th component C,i whose abscissae are 



the same as those of j'p, j'j, jt'^, Vn-i respectively, then 



Ji'o - Ji'i +yi - ■ • +Jn-1 = «No=: - ;/Nii=;/No = = fl^n-l 

 when « is an odd number, and 



jo-ji+y-i- • ■ ■ -yn^i=--0 



when n is an even number, as we are now consideriug odd 

 periodic functions only. 



Thus from ;/ e.s. ordinates of the original half wave we obtain 

 only one ordinate per half wave of Cn , so that in order to obtain 

 ;// e.s. ordinates per half wave of C„ it is necessary to have mn 

 e.s. ordinates of the original half wave. 



For instance, to obtain 3 e.s. ordinates of C,i we must measure 

 3n e.s. ordinat(-s oi g{x). Let these be 



Joi J'n J''l .V-M Jill-li 



and let the corresponding ordinates of C,i be 



No, N„ N„ N„ N3„-i, 



then 



yo—y^+Jr.- ■ ■ +Jl'3<i-3=«No=-«N3 = «N,;= =«N3„_3 

 yi—yi+yi— ■ ■ +y-sn^2 = f'^i=-"'^i="^i= =fl'^3n-2 

 yi—yr.+yn- • ■ +y3n-\ = »^-2=-ft^i = ^''^H= =/t^3n-l 



Subtracting now the ordinates of C„ so obtained from the cor- 

 responding y ordinates, we obtain a new set of 3« e.s. ordinates 

 which are those of the original half wave with its «th component 

 removed. 



5. In practice it will generally be sufficient to determine the 

 1st, 3rd, 5th, 7th and 9th harmonics (Hj H3 H- H^ H,, say). 

 This can be done with considerable accuracy when 15 e.s. ordin- 

 ates of the original half wave are given. 



Thus if these be 



J'O) J'l) J'-2) • • • J'l4 



corresponding to the angular abscissae 



*-0) •^15 -^2) • • • "^14 



v,'here Xi- Xq = X2- Xi= . . . =Xii- Xy^=:Tr/l5, 



and if s^, z^, z.^, z.„ s^, be 5 e.s. ordinates of the half wave of 



