Harmonic Analysis. 401 



given in row 5 of Table I. Every figure necessary in the calcu- 

 lation will be given. 



The fii'st row of figures in Table I. are the abscissae Xq, x^, etc. 

 to which the given ordinates correspond. Space for three rows 

 of figures is left, and then the 15 given ordinates are written 

 down. The.se are divided into three sets of five each, and the 

 numbers of the middle set are subtracted in order from the sums 

 of first and last set, giving five numbers which are the corres_ 

 ponding ordidates of 30,. Space for two or more rows is left 

 and the given ordinates are now written down as in the table, in 

 two rows of six each and one row of three, in order. The 

 columns formed are added and the la.st three of the sums are 

 subtracted from the first three, giving three ordinates of 50,. 

 The first of these minus the second, plus the third, gives one 

 ordinate of ISCj.,, whose other ordinates are got by alternating 

 the sign. Subtracting 50^ from 5C,- we obtain 5H-. Having 

 obtained C^^ we now subtract SCi,, from 30.^ and olitain 3( Hh+ Hd)- 



Above the given ordinates write those of C.^ with signs cliangecl 

 (row 4), and above these write those of H,- with signs changed 

 (row 3). Add rows 3, 4 and 5 to get row 2, in which are the 

 ordinates of Hj + H, + Hu etc. Neglecting H^, Hj,, etc., as is done 

 the analysis in Table I., we may consider the figures in row 2 as 

 the ordinates of H,, and neglecting H,, we may consider the 

 figures in row 11 as the ordinates of 3H,,. 



The first 15 numbers under Amp. Hj are the quarter squares 

 of the ordinates of Hj. Twice the sum of these is divided by 15, 

 the number of ordinates, and the quotient is found to be the 

 quarter square of 987. Hence //j, the amplitude of Hj, is 987. 

 Similarly for the amplitudes of Hg and Hr,. 



Under the heading "phase of Hj," in the first colunni under 

 sines, are the quotients got by dividing the first four ordinates 

 on the rising side of H^ and the last four on the falling side of H^ 

 by /i^; in the second column under angles are the corresponding 

 angles, and in the third column are the eight values of 12° — a 

 deduced. The mean of these 2° 2' when subtracted from 12" 

 gives the crossing point or phase of Hj as 9" 58'. Similarly for 

 the phases of H^ and H,. It will be noticed that at the crossing 

 point determined for H.^, H^ crosses down, which is expressed 

 analytically by writing its amplitude negative. 



