Studies on the Motion of Viscous Flows — III 



V"' 



F (x) = — + ) ( Aj^ cos nx + B^ sin nx) , 



then 



7T a rt CO 



IJ f(x)F(x)dx= ^+ 2 (aA + b„BJ 



(2.14) 



Theorem III [41]: Using Perseval's Theorem, we can show that 

 if the product 



f (X) • F(x) 



'o v"" 



— + ) ( a^ cos nx + bj_| sin nx) 



+ 2, ( A|^ cos nx + B^ sin nx) 



(2.15) 



is represented by the Fourier series 



f (x) • F(x) = — + 2_, (a^cosnx+ /3^ sinnx) , (2.16) 



then the coefficients a^, a^, and /?„ are given by the following 

 expressions: 



^o\ 



+ Y (ar.A„ + bB ) 

 / . ^ n n n n ' 



a„B 



2 2 



B 1 



n 1 



+ - 



2 2 



+ \ H [3m(A,+ n+ Vn) + b^(Vn+ Vn)] (2.17) 





with the stipulation that 



B_,. ^ -B,. . 



(2.18) 



Using Eqs. (1.18) and (1.19) and Theorems II and HI, we obtain from 

 Eq. (1.54) 



559 



