342 WAVE ANALYSIS. [Kxr. 



A. = 1/9 ( j/ sin 30 + s 2 ' sin 60 + j/ sin 90 ) , 



which is the form used in the Table, 8. 



16. A 3 is developed in terms of sin k 30 while A u is developed in 

 terms of sin 150. The latter can be written sin(i8o 30), 

 which expanded gives sin k 180 cos k 30 cos k 180 sin k 30. It 

 is evident that, when k is odd, sin k 30 = sin k 150, while when k is 

 even sin k 30 = sin k 150; hence, A 1S can be determined from A 9 

 by changing the signs of all terms in the third column of equation (8). 

 Hence 



A u = i/9 [>/ sin 30 j/ sin 60 + V sin 90]. 



The other coefficients are determined in a similar manner. 

 17. Check. In equation (5), y = y a when ^ = 90; and y = y 

 when x = o. Substituting these values, we have 



After analyzing a wave, these equations may be used to check the 

 values of A v A 3 , A 6 , etc., and B 19 B 3 , B 6 , etc. 



APPENDIX II. 



ANALYSIS OF WAVE WHICH MAY HAVE EVEN AND ODD HAR- 

 MONICS AND A CONSTANT TERM. 



18. Method of 12 Ordinates. In the most general case,* when the 

 negative part of the wave is not equal to the positive and is not a 

 repetition of it, both even and odd harmonics may be present and also 

 a constant term, B . To analyze such a wave, see Fig. 3, equidistant 

 ordinates must be taken over an entire period, or 360, and not merely 

 for a half period, as in the preceding pages when odd harmonics only 

 were considered. 



Let ordinates be takenf at intervals of 30, i. e., there are 12 known 

 ordinates in a complete wave. 



* This general method, which applies whether there is a constant term or 

 not and whether odd or even harmonics are present or not, is taken directly 

 from Runge's article,, where the method is given in detail for 12 and for 36 

 ordinates. 



fFor greater accuracy ( 12) more ordinates must be used. 



