50 



93. The demonstration of Fourier's series shows that if y is any 

 function of x, its values between the hmits of a:=0 and x=l are given 

 by the series: 



y=Boi-B, cos {Trx/l) + Ci sin (xx/O+^a cos (27rx/Z)+(72 sin (2tx!1) 

 +^3 cos (3 xj-//) + 03 sin {3wx/l)+ . . . (49) 



in which: 

 Bo 

 B 



{llT)^'^ydx, 



j=(2/Z) y cos (Txll)dx, Ci=(2ll) \ y sin {Trxfl)dx, 



B2^(2/l) I y cos i2Trxll)dx, €2= (2/1) y sin {2irxll)dx, 

 Jo Jo 



B^= (2/1) j ?/ cos (3Trx/l)dx, (73= (2//) ) ' y sin (3Tx/l)dx, (50) 



and so on. The angles in these equations are expressed in radians. 



94. If T is the length (in mean solar hours) of the component day, 

 then since 2t radians=360°, 27r/T=a, the speed of the diurnal com- 

 ponent. 



Placing x=t and l=T, equation (49) becomes: 



y=BQ-\-Bi cos }{at^Ci sin ){at-{-B2 cos a^+C'2 sin at-^B^ cos 3/2 at 

 -\-C3sm3/2 at +Bi cos 2at-\-Ci sin 2at+ . . . (51) 



Equation (51) is the development of any function of t. It becomes 

 the development of the particular function of t expressed by equation 

 (46) if it is identical with the latter, i. e., if the coefficients of the identi- 

 cal terms in the two equations are equal, the coefficients of the terms 

 in equation (51) not appearing in equation (46) becoming zero. It 

 follows therefore that 



H,^B,= (l/T)^'^ ydt, 



rr rr 



Ci—B2—(2/T)\ y cos at dt, Si = C2= \ y sin at dt, 



C2=Bi= (2/T) ?/ cos 2 at dt, s2=Ci= \ y sin 2 at dt, etc. (52) 



An examination of the form of the integrals in equations (52) dis- 

 closes that (1/T) I ydt is the mean value of y between the limits of 



