DETERMINATION OF WAVE FORM 653 



only the odd harmonics are present, and consequently the values 

 of k are odd numbers. It is necessary, therefore, to deal with 

 only one-half of the wave, and 2n spaces per half wave are used, 

 making A0 = ^~- 



The method of grouping the terms so as to economize time 

 may be explained as follows. Referring to (13), 2/1 is multiplied 



A TT Jc -JT 



by sin x- and 2/2n-i by sin (2n - 1) . As k is odd, 



and in general, 



/r , N kir . mkw 



sm (2n -m}-^ n = sin -^ (17) 



Consequently, 



1. The number of multiplications may be halved by adding, 

 before taking the products, those values of y for which the sum of 

 the subscripts is 2n. 



2. Again, the same products are needed in A k and A 2 n-*; 

 that is, in those coefficients for which the sum of the subscripts 

 is 2n, since 



sin (2n - k) ^ = sin fc g (18) 



If the number of the ordinate concerned in the multiplication 

 (m) is even, the sign of the left hand member is -, and if m is 

 odd the sign is +. 



For example, suppose a half period is divided into 2n = 12 



equal parts (&6 = - =c= 15 ) and the ordinates measured: 



then by (13) and (17) 



A i = H [(2/1 + Vii) sin 15 + (2/2 + 2/io) sin 30 + 



(2/3 + 2/9) sin 45 + (y, + 2/s) sin 60 + (y* + 2/?) sin 75 + 2/6sin90] 



4 11= % [(2/1 + 2/n) sin 165 + (2/2 + 2/io) sin 330 + 



(2/3 + 2/9) sin 495 + (2/4 + 2/s) sin 660 + (2/5 + !/?) sin 825 + 



2/6 sin 990]. 



For convenience the sines of all the angles maybe expressed 

 in terms of the sines of angles of 90 or less. 



