ii-A] WAVE ANALYSIS. 341 



14. Proof. The foregoing expressions may be derived* as 

 follows. To determine a coefficient A n , multiply the first of the 18 

 equations by sin o, the second by sin 10, the third by sin 2n 10, etc., 

 and add. The sumf of all terms that contain A n on the right hand of 

 the equations (after multiplying) is gA n ; the sum of the other terms 

 is zero. The sum of the left hand terms may be written as a summa- 

 tion. Thus, 



2 ^sin nk 10 = gA n . (7) 



Transposing, we have the value of A n , as in (6). The value of B n is 

 similarly found by multiplying by cosines instead of sines. 



15. Determining Values for Individual Coefficients. Given the 

 general expression (6) for A n and B n , the next step is to find parti- 

 cular expressions for A 3 , A 5 , A 7 , etc., which may be conveniently used 

 in a numerical solution. It will suffice to determine A 3 as an illustra- 

 tion. In (6) or (7), let w 3 and assign values for k from o to 17. 

 We then have 



9^s = 4- y sin o -f- y^ sin 30 + y z sin 60 + y 3 sin 90" 

 4- y e sin 180 4- y s sin 150 + y* sin 120 



4- y 7 sin 210 4~ y 8 sin 240 -\- y 9 sin 270 

 4- y 12 sin 360 -f- y u sin 330 + y 10 sin 300 



-\- y 13 sin 30 4~ 3^4 sin 60 ~h ^Vis sin 90 



4- y 17 sin 150 -f ;y 18 sin 120 



Since sin 150 = sin 30, sin 210= sin 30, sin 330 = sin 30, 

 sin 120 = sin 60, sin 240= sin 60, sin 300 = sin 60, sin 

 270= sin 90, we may write (8) as follows: 



9^3 = (yi + yn 4- y, 4- y a y, y u ) sin 30 



+ (y 2 -f- ^i + ^ + y u ^s y M ) s ^ n ^ 



y 9 ) sin 90 



*(i4a). General Expression for Coefficients. Determined more gen- 

 erally for an infinite number of terms, the coefficients of the nth order are 



| y sin nx dx\ B n = I y cos nx dx. 



(See Byerly's Fourier's Series and Spherical Harmonics, Chap. II.; Tod- 

 hunter's Int. Calculus, Chap. XIII. ; Greenhill's Int. Calculus, 183 ; etc.) 



t If there are ra terms and ra equations instead of 18, this sum is 

 the average value of the sine of an angle, squared, being y 2 . 



