ANALYSIS OF CURVES. 433 



= a a t sin 9 a. 2 sin 2 6 a s sin 30 .... 



+ 6, cos + & 2 cos 2 + 6 3 cos 3 + . . . . 



.-. Yt+ Y. 2 = 2 (a + &! cos + 6 2 cos 2 + 6 3 cos 30 + . .) 



y t - F 2 = 2 (a, sin + a, sin 2 + a, sin 3 6 + . . . .) 



Y -\-Y 



Thus - 2 will consist of the cosine terms of the 



u 



Y Y 



original expression alone, and - 2 will be the sine 



a 



terms 



Method of Procedure. Plot the curve to a large scale, a& 

 for the previous method of analysis. 



Draw up a table (see Table III.) with the angles 

 showing the sub-divisions which it is proposed to take in 

 the first column. In the second column enter the measured 

 values of the curve corresponding to these values. In 

 column 3 put the values of the curve for the angle (360 - 

 angle in first column). In column 4 enter the difference 

 between the numbers in columns 2 and 3. Column 5 is 

 obtained by dividing the figures in column 4 by two. 

 Column 6 is the sum of the readings entered in columns 

 2 and 3, while column 7 is the half values of these 

 numbers. 



The coefficients of the cosine terms will be obtained 

 from column 7, whjle those of the sine terms will be got 

 from column 5. 



In order to determine the coefficients for the wth 

 harmonic proceed as follows : 



For the sine term, divide the complete period of the 



curve into n parts, taking the angle ^- as the starting 



An 



point, since this makes sin = 1. 



Measure the ordinates of the curve for the angles 



H- + <Y- H Add these ordinates together 

 2n, 2n n , 2w n . 



and divide by n. The result is the required coefficient. 



For the cosine term, divide the curve into n parts, 

 starting from the angle (making cos = 1). 



Measure the ordinates for the angle 0, , - Add 



them together and divide by n. 



Having obtained one harmonic, e.g., the third, subtract 

 its corresponding ordinates from the original curve, and 

 so obtain a new curve which is the original curve third 

 harmonic. 



