The arbitrary function can be defined in terms of F' and b'/ in the following
manner,
6
F' = Fk(kx+ Wk)
k= 1
6
b' = (,,x +Pky) (A.41)
k= 1
Now from equation a and b
Sx + k) ) ,(x + UkY) + ak(x + UkY) + bk (A.42)
12 (11k)
where ak and bk are arbitrary constants.
Now there are four theorems, which are important for the theory of the
equilibrium of a body bounded by a cylindrical surface.
Theorem 1: The equation 14(/) = 0 cannot have real roots.
Theorem 2: The equation 12(P) = 0 cannot have real roots.
Theorem 3: The equation 14()12(p) l = 0 cannot have real roots.
Theorem 4: The equation anip4 2a16/3 + (2a12 + a66)/2 2a26 + 22 = 0
cannot have real roots.
Based on the third theorem, we can show that the numbers are al-iv
complex or purely imaginary.
The general expression for the stress function can now be shown as,
F 2Re[Fi(z) + F2(z2)+F3(z3)]+Fo0
=y 2Re[AiF(zi) + A2F(z2)+ F 3) ','+ (A.43)
A3 -) +(.3