340 WAVE ANALYSIS. [Exp! 



(m i) harmonic, assuming that higher harmonics are negligible; 

 thus, 18 ordinates determine harmonics to the seventeenth, 6 ordinates 

 determine harmonics to the fifth, etc., assuming that no higher har- 

 monics are present to an appreciable extent. 



Reducing the number of ordinates used, however, not only reduces 

 the number of harmonics that can be determined but reduces the 

 accuracy with which these are determined. For example, if the 

 seventh or ninth harmonic has considerable amplitude, the method 

 with 6 ordinates in general would not accurately determine even the 

 third and fifth harmonics. The more ordinates used, therefore, the 

 more accurate is the method, but the labor is likewise increased. 



Except in work of the greatest precision, the use of more than 18 

 ordinates in a half-wave is hardly worth while, for the accuracy of 

 the data will rarely warrant it. On the other hand, it does not pay 

 to make an analysis with too few ordinates and to risk errors in the 

 result, unless only an approximate analysis is desired. 



For simplicity, the following discussion is limited to the method 

 using 18 ordinates, but it can be readily modified so as to apply when 

 more ordinates, or less, than 18 are used. 



13. Development of Method. For practical use the infinite series 

 in equation (i) must be limited to a finite number of terms; thus, ex- 

 cluding all harmonics above the seventeenth, we have 



y = A l sin x -f- A 3 sin $x -\- A 5 sin $x - A 17 sin ijx 



-f- B l cos x -\- B 3 cos 3* -|- B 6 cos $x - J5 17 cos 17*. (5) 



Substituting for x 18 known consecutive values (o, 10, 20, etc.) and 

 for y the corresponding 18 known values (y , y lf y v etc.), we have 18 

 simultaneous equations of the first degree which may be solved for the 

 18 unknown coefficients A l to A 17 and B^ to B ir 



The coefficients of the wth harmonic may be written as summations, 

 in which the values of k vary from o to 17; thus 



J 8 *=o * 



2 *= 17 



