Harmonic Analysis. 399 



Hj + H- obtained above, and from the smooth curve drawn 

 through them to measure off 14 e.s. ordinates. From these, 2 

 e.s. ordinates of the half wave of H,, and which determine H,, 

 can be obtained. By subtracting these from the corresponding 

 ones of Hj + H^, 14 corrected ordinates of Hj are obtained. 



6. It now remains to determine the amplitudes and phases of 

 the harmonics of Ci from their ordinates which we have obtained. 

 It is easy to show that 



?|sin^^ + sin^(^ + -) + sin^(^ + — ) + • . 



+ sin^(^ + ^^l^)} = l, 



from which we conclude that the square root of twice the mean 

 of the squares of n e.s. ordinates of half a sine wave is equal to 

 its amplitude. 



Hence, with the help of a table of squares or of the quarter 

 squares given in most sets of tables the amplitudes of H^ H3, 

 etc., can be quickly determined. 



[The rule that the amplitude is equal to 7r/2 x mean of the 

 ordinates is only sufficiently accurate when a large number of 

 ordinates is taken.] 



If rto) ^^ly '^2) • • • ''u be the ordinates we have found for 

 Hi corresponding to the angular abscissae Xq, x^^ x., . . . x\^ 

 respectively, and if //j, a be the amplitude and phase of Hj or in 

 other words, if 



Hi = ;^isin( wt — a), 

 then any of the equations 



sin(.x:o - a) = aj/ii 



sin(.a;i - a)^^ajh-^ 



&\u(x^ — a)^a^lhi etc., 

 would determine a, provided the ordinates rtg, ^j, a.,, etc., are 

 exactly those of a sine wave. 



In practice, however, small upper harmonics will invariably be 

 left in flo, ^1, aj, etc. [it may not have been thought worth while 

 to remove H7], and though their amplitudes may be negligably 

 small, yet they might cause considerable error in the value of a 

 when determined from only one of the above equations. Hence it 



