Figure 170a Figure 170b 



Figure 170 — Two sector-shaped cylindrical profiles. See Section 102. 



Equation [101a, b] a Fourier series composed of terms of the form of Equation [101i,j] and 

 giving suitable values to the coefficient A^. 

 Assume that 



1 , sin 2 6* , 

 -— cor - (ua 



2 cos 2 a 



z 



,(2n + 1) 



(2n + l)- 



, [102a] 



where 6 is measured as before from the bisector of the apical angle. Then, to make 

 q^ = (- dtji/dr) = at r = a. 



= oja 



sin 2 9 V~~i 77 



+ oia / (271 + 1) /lo„ , 1 sin 



cos 2a Z__i ^ 2a 2^+^ 



77(9 



(271+1) — 



To find ^2/c + 1' "multiply through by sin [{2k + 1) T76/2a\ and integrate from 6 = -a io =<t. 

 It is found that, replacing k by n. 



^2. + i =(-!)" 



32 a 



77 (2/1 + 1) [(271+ 1)2 77^ -IGa^] 



[102b] 



The complete Fourier series would include also terms containing cos [(27i+l)770/2a], but if 

 these are included their coefficients are at once seen to be zero because of symmetry. 

 The corresponding expression for ip \s 



1 2 cos 2 6 



l/f = — CJ T 



2 cos 2a 



L 



(2n + 1 



) -^ 







2a 



77 61 





COS 



(271+1) 



' 2a 



. [102c] 



251 



