Computation of certain Harmonic Components. 



67 



The quadrant to which the angle T belongs will depend on the 

 algebraical signs of the coefficients p q, and the following table (in 

 which t is the angle corresponding to -f p, + q) shows the cases that 

 may arise. In it are also indicated the positions of the earliest 

 maxima, of the several harmonic components. 



Coefficients. 



Value 

 and position 

 of T. 



Position of earliest maximum of components. 



First order. 



Second order. 



Third order. 



Fourth order. 



+p 



+ <Z 



T = t 

 0° to 90° 



/ii = 90°- t 

 0° to 90° 



ju 9 = 45° — it 

 0° to 45° 



^3 = 30°-^ 

 0° to 30° 



/i 4 =22*°- it 

 0° to 22i° 



-P 



+ 2 



T = 360-* 

 270° to 360° 



H = 90° + t 

 90° to 180° 



,« 2 = 45 + f* 

 45° to 90° 



^ 3 = 30 o + ^ 

 30° to 60° 



^ 4 = 22^+ |f 

 22|° o45° 



-v 



~1 



T = 180 + t 

 180° to 270° 



^ = 270°-* 

 180° to 270° 



ft = 135°-^ 

 90° to 135° 



p 3 -- 90° 

 60° to 90° 



/«4= <o1\°-kt 

 45° to 67^° 





-1 



T = 180- * 

 90° to 180° 



& = 270° + t 

 270° to 360° 



M = 135° +y 

 135° to 180° 



^ = 90° + \t 

 90° to 120° 



671° to 90° 



7. 



The foregoing discussion assumes that the series of quantities dealt 

 with is truly recurrent, that is to say, that the 25th observation will 

 be exactly coincident with the 1st, or a — a u . In fact this will rarely 

 be the case, and it becomes necessary to ascertain what effect any 

 non-periodic change, or want of coincidence between the beginning 

 and end of the series will have on the values of the several p q 

 coefficients. 



In the absence of any knowledge of the law which determines such 

 a non-periodic change, it may be assumed to be uniform for the 

 period over which the observations extend. Further it will be con- 

 venient to refer the change to the middle of the period, or to the mean 

 value of the series with reference to which the periodic variations are 

 being considered, so that the non-periodic deviations will be equal 

 and affected by opposite signs, at equal intervals on either side of the 

 middle of the period. 



Hence assuming that 2c is the whole non-periodic change with its 

 proper sign, so that a + 2c = a 24 , the correction of any observation 

 a n in order to eliminate the non-periodic change which affects it, will 



? 2 



