. Graphical and mechanical computation . 3 n, . . kir\ = n when k = 2n, An, 6n, . .= wcos—i , , n [ = — n when k = n, 3 n, 5 n, . . ^?Vsin kxT = ^sinf & —\- kr —-] = o for all values of k. ^3/ = na0 — nan + wa2 n — wa3 n + • • • = w (a0 — o„ + a2n - as n + ain — • • • ).Subtracting the second set of ordinates, yf, from the first set, y, wehave %yr- 2,yrf = y£(yr-y/)=yo-yo-\-yi-yi+y2-y2+ • • • +yn-i-yn-i = 271 (a„ + fl3n + a5n + • • • ), or an+a3n+a5n+ = — bo-yo+yi-yi + -\-yn-x — yn-i). The first set of n ordinates start at x = o and are at intervals of 2 ir/n,and the second set of n ordinates,
. Graphical and mechanical computation . 3 n, . . kir\ = n when k = 2n, An, 6n, . .= wcos—i , , n [ = — n when k = n, 3 n, 5 n, . . ^?Vsin kxT = ^sinf & —\- kr —-] = o for all values of k. ^3/ = na0 — nan + wa2 n — wa3 n + • • • = w (a0 — o„ + a2n - as n + ain — • • • ).Subtracting the second set of ordinates, yf, from the first set, y, wehave %yr- 2,yrf = y£(yr-y/)=yo-yo-\-yi-yi+y2-y2+ • • • +yn-i-yn-i = 271 (a„ + fl3n + a5n + • • • ), or an+a3n+a5n+ = — bo-yo+yi-yi + -\-yn-x — yn-i). The first set of n ordinates start at x = o and are at intervals of 2 ir/n,and the second set of n ordinates, start at x = ir/n and are at intervals of2 ir/n; thus, the period from x = o to x = 2 ir is divided into 2 w equalparts each of width ir/n (Fig. 91a). Hence, If, starting at x = o, we measure 2 n ordinates at intervals of -w/n, theaverage of these ordinates taken alternately plus and minus is equal to themm of the amplitudes of the nth, 3 nth, 5 nth, . . cosine Fig. 91a. Thus, to determine the sum of the amplitudes of the 5th, 15th, 25th,. cosine components, merely average the 10 ordinates, taken alter-nately plus and minus, at intervals of 1800 -7- 5 = 360, or at o°, 360, 720,. . , 3240 (Fig. 91c); therefore G5 + ai6 + a25 + * = tV (yo — ^36 + yn — yios + ym — ym +ym — ^252 + ^288 — ^324). Art. 91 NUMERICAL EVALATUION OF THE COEFFICIENTS 195 If the 15th, 25th, . . harmonics are not present, then £5 = A (yo — y™ + ^72 — ^ios + 3*144 — ym + ym — y™ + y^ — y*i\)> 2x (3) Similarly, if we start our intervals at x0 = —?, then xr = —— + r —KJ/ J 2 11 2 n n and 2}cos kxr = 2)cos Ik — + kr —J = o for all values of k, ^sin kxT = 2)sin Ik — + kr —\ = o except when k = n, 2 n, 3 n, . kir wsin 2 n = 11 when k — n, 5 n, 9 n,= o when k = 2 n, 4 n, 6 n,= — n when k = 3 n, 7 n, 11 n, .. ^yr = nao-\-nbn — nb3n-\-nbbn— • • • = n (a0+6„ — &3„+&0n — &?«+ • • • )?
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