88 THE STUDY OF SPEECH CURVES. 



period {T=mh), beginning atO and ending at (m~l)h, then the values ^o<^ 

 kit, hit, . . . become 0, h, 2h, . . . (m~l)h. The set of equations then 

 becomes 



?/o = C+Oi.cos +O2.COSO + . . . +6,.sinO +&2.sinO + . . . 

 yi — C+ai.cosh + aj.cos 2/i + . . . +&,.sin^ + fcj-sin 2/i + . . , 

 y.i — C+ai.cos2h + tti-cos 4h + . . . +61. sin 2h + 62. sin 4h + . . . 



The constants C, a„ a., ... 6,, 62, . . . can be most advantageously- 

 found by the method of least squares; this requires that the sum of the 

 squares of the residual errors be a minimum, or 



(— i/„+C+ai.cosm/i.+a2.cos 2mh+. . . +bi.smmh + h2. sin 2mh+. . .)'^=min. 



Differentiating this sum in respect to each of the constants and 

 putting the result = 0, we obtain the set of values given in equation (21) 

 above. 



The labor of calculating the coefficients is greatly diminished by 

 making m a factor of 360; the sines and cosines then repeat themselves 

 in simple relations and the additions can proceed according to prepared 

 schedules. The method of making and testing the schedules can be illus- 

 trated with the case of 12 ordinates. In the equations (21) m =12, /i = 

 30". The value C= h of the sum of all the ordinates (+ and — not 

 forgotten). For the first coefficient we have 



ai = i{yo.cosO° + yi.cos 30° + ?/2-cos 60°+2/3.cos 90° + i/4.cos 120° + i/5.cos 150°+ 

 2/6.C03 180'' + 2/7.cos 210^ +2/3.003 240°+2/9.cos 270°+i/,o-cos 300° + j/n- cos 330°). 



But 



cos 0°= 1.00, cos 180°= —1.00, 



cos 30°= cos 330°= 0.87, cos 150°= cos 210°= -0.87, 

 cos 60°= cos 300°= 0.50, cos 120°= cos 240°= — 0.50, 

 cos 90°= cos 270°= 0. 



Thus, we have 



ai = ^ (2/0 + 0.872/. + O.5O2/2 + - O.5O2/4 - 0.872/5 - 2/6-0.872/, - 0.50y, 



-0 + 0.502/10 + 0.872/u). 



