HARMONIC ANALYSIS. 89 



We obtain likewise 



61 =i (0 + 0.50y, + 0.87y, + y, + 0.87y, + 0.50y, + -0.50y, -O.STys - y^ 



- 0.87j/.o-0.50j/„), 



a, = i(yo + O.SOt/, -0.50,-^3-0.50^4 + 0.50y, + ye+0.50y^—0.mys — y^ 



-0.502/.o + 0.50yn), 



6» = i(t/o + 0.87y, + 0.871/j + -0.87y, - O.S7y, - 0« - 0.87y, + 0.87 j/, + 0,, 



-0.871/10- 0.87 j/u), 



and so forth. 



For the entire set of coefficients it is necessary only to multiply each 

 ordinate by 1.00, 0.87, 0.50, 0, and select the requisite values with the 

 proper signs. The formulas show that the selection is to be made accord- 

 ing to definite rules. When the values are written in 4 columns of 12 

 lines (see the schedules at end of volume) the selection for a always begins 

 with + in the upper lefthand corner, and for b with + in the upper right- 

 hand corner. For a, it proceeds continuously "1 to the right and down, 1 

 to the right and down," etc.; for a,, "2 to the right and down, 2 to the 

 right and down," etc. For 6, it proceeds " 1 to the left and down, 1 to the 

 left and down," etc.; for b-i, "2 to the left and down," etc. Whenever 

 the count strikes the righthand edge, the sign changes from + to — or 

 from — to +. 



The labor of selecting the values according to these rules is largely 

 avoided by using patterns that cover all except one set at a time, and at 

 the same time indicate the signs. Thus a pattern is made for a„ another 

 for Gi and so forth; the patterns for 6i and 6j . . . are obtained by turn- 

 ing those for the corresponding a's over and marking the signs properly. 

 Thus for 12 ordinates there will be 6 patterns, but each one is used on 

 both sides. For 24 ordinates there will be 12 patterns, etc. It will be 

 found that for the last b in each set the pattern indicates a sum of zeros ; 

 this can therefore be omitted. The corresponding pattern for the last a 

 can also be omitted, if it is remembered that for this a the values of 

 the first column are taken with alternate + and — . The simple zigzags 

 for Oi and h are also readily memorized. Thus for 12 ordinates only 4 

 double-sided patterns are absolutely necessary. 



At the end of this book the schedules for 12, 24, 36, and 72 ordinates 

 are given. The labor of preparing such schedules is very great and the 

 chances of mistakes are numerous. All the patterns given in this volume 

 have been computed by at least three persons independently. They have 

 also been subjected to a series of tests sufficiently detailed, it is hoped, 



