Harmonic Analysis. 403 



and C,5. To determine Cj, replotting will have to be resorted to 

 if the full wave trace be not available. 



At the top of Table III. are written the 24 given ordinates 

 under their corresponding abscissae. From these ordinates the 

 constant terni of J (/) has been removed. This can be done by 

 aid of the formula 



/(0+/(< + ^/«)+./"(^+2t/«)+ . . . +/{t+,r-lr/H) 

 — f/[ao + an sin«(u>/ — 9n ) + a-2n^n\'2n{(i>i—6-2n) 



-f rt;3„sin3«(aj/— ^3„) + etc.] (Ill-) 



which can be easily established by the method used in § 2. 



From this formula we see that the mean of n e.s. ordinates 

 embracing one period of a periodic function is equal to its con- 

 stant term, if its «th, 2«th, etc., harmonics are neglected. 



Returning to Table ill., we add the second twelve ordinates 

 with their signs changed to the iirst twelve, in order, and obtain 

 12 e.s. ordinates of 2Ci, i.e., of 2[Hi-|- H8 + Hj + ]. (See equa- 

 tion I., § 2). 



Subtracting these from twice the given ordinates, those of 

 2[H2-f H4-t- Hg-f ] are left, and the remainder of the work 

 proceeds as in Table I. 



2[H2 4- H4-f H,; -I- etc.] could be obtained directly from the 24 

 given ordinates by adding the second 12 to the first 12 of them, 

 in order. (See formula III., § 9). 



The amplitudes and phases of the different harmonics were 

 determined as in Table I., but the figures necessary in their 

 calculation are not given. 



The following are interesting applications of the above method 

 to more general harmonic analysis. 



10. To obtain the harmonic expression for the odd periodic 

 function whose graph for half a period is the sides of an isosceles 

 triangle of altitude h. See Fig. 1. 



Taking o and ir as the abscissae of the extremities of the base, 

 relative values of any number of e.s. ordinates can be written 

 down, and any component at once obtained. Thus, 30 e.s. ordin- 

 ates would be 0, 1, 2, 3, . . 14, 15, 14, . . 2, 1, and these 

 correspond to an altitude 15. 



