432 ANALYSIS OF CURVES. 



Let n = 3 ; then since must be an integer, all the 



W 



terms will cancel except those for which m is a multiple 

 of 3, i.e., all harmonics will vanish except the third, sixth, 

 ninth, &c., and 



2/i + */-. + 2/3=3 a, sin 3 (8 + a s ) + 3 a c sin 6 (6 + a) 

 + 3 a,, sin 9 (0 + a 9 ) + . . . 



If the curve is symmetrical, even harmonics will not 

 exist, hence the term containing sin 6 (0 + o 6 ) may be 

 neglected. Further harmonics above the fifth are seldom 

 present of sufficient amplitude to be worth determination. 

 Hence we have for practical purposes 



2/i + y s + ft = 3 a 3 sin 3(0 + o s ). 



By choosing the value of such that 3 (0 + a 3 ) = 90, 

 we can determine a 3 from observed values of y l3 y.>, 

 and y z . The method of doing this is, then, as follows : 



From the curve, measure the value of the ordiuate 

 qa 



for the angle 6 ^ = 30. Call this y,. 

 w 



2:r 



Similarly measure the ordinate at B -\ = 30+120= 150 



72- 



Call this t/2 

 also the ordinate at #+ = 30+240 = 270 



Yl 



Call this y 3 



Add the three measured ordinates together and divide by 

 n = 3. The result is the coefficient of sin n B = sin 3 0. 

 Similarly by starting from the angle proceed to obtain 

 the coefficient of cos n 6 = cos 3 B. 



The analysis may be further simplified by separating 

 sine and cosine terms, as follows : 



Let 



/ (0) = y t = a + fl^ sin B -f o 2 sin 2 B +a 3 sin 3 B + a 4 sin 4 B + 



+ 6, cos + 6 2 cos 2 B +6 3 cos 3 +6 4 cos 4 + 



/ (2ir -0) = r, = o +a, sin (2 TT- 0)+a. 2 sin 2 (2 TT -0) +a 3 sin 3 

 (2ir-0)+ . . . 



+&! cos (2 TT - 0) + 6 cos 2 (2 7T- 0) + & 3 cos 3 

 (2 TT - 0) + . . 



