Reply to the remarks of Golecki

Brief Communications

It is shown that in this case using the stress function

#=AAr2+Blnr

(A, B constants)

(4)

it is possible only to satisfy the equilibrium equations, de Saint Venant’s compatibility equations and boundary conditions, but it is not enough to obtain the single-valued displacement field, as the region under consideration is multiplyconnected. The additional condition must be satisfied in this case [compare (1)) p. 42431, which secures the single-valued determination of the displacement field. Using the stress function in the form (4) it is possible to satisfy this condition when G = const (the body is homogeneous), but not when G = Go( l/r2). This is all that was considered in (1). The single-valued solution of this problem was presented in 1959 in (2), p. 393396, in the formt2 % = Br-‘,

up = 0) ur = C - D(R&)*,

mB= C + 3D(R&)*,

7,@= TqP= 0)

(5)

where

c=

D=

-Pl+l

B=EL4

PI -

Pz

Go 1 - (RIIRz)~'

Unfortunately Chen was not acquainted with this last paper and also rederived in 1963, this solution (compare p. 93, Case III, formulas 30-33). Finally, it is worth pointing out that the investigations on plane stress of a two-dimensional incompressible, non-homogeneous (isotropic and linear elastic) body were presented in 1959, in (3). References

(1) J. Golecki, “On the Foundation of the Theory of Elasticity of Plane Incompressible

Non-homogeneous Bodies,” in, Non-homogeneity in Elasticity and Plasticity Proc. International Union of Theoretical and Appl. Mech. Symp., Warsaw, Sept. 2-9, 1858, New York, Pergamon Press, Inc., 1959. (2) J. Golecki, “On the Foundation of the Theory of Elasticity of Plane Incompressible Nonhomogeneous Bodies,” Arch. Mech. Stos., Arch. Mecan. Appl., Vol. 11, No. 3, 1959. (3) J. Golecki, “The Stress Function for a Two-dimensional Incompressible Non-homogeneous Body in the Case of Plane Stress,” Bull. Acad. Polonaise SC. Se& Sri. Tech., Polska Akad. Nauk., Warsaw, Vol. 7, No. 6, 1959.

Reply to the Remarks of Golecki btj YU MEN

Department of Mechanics Rutgers, The State University, New Brunswick, New Jersey We would like to thank Professor Golecki for pointing out that the singlevalued solution for the “inverse-square” case was actually given by him earlier * The work reported in (2) is a further development of the investigation described in (1).

Vol. 2136, No. 3, September

1968

257

Brief Communications

in Arch. Mech. Xtos., Arch. Mecan. Appl., Vol. 11, 3, 1959. Since this said journal was not widely available in this country, the solution to the same problem was redrived. It is now discovered that the author’s solution checks identically with that of Professor Golecki.

A Counterexam$e b?J

to a Generalized Ergodic

Theorem

T. J. WAGNER

Depatiment of Electrical Engineering University of Texas, Austin, Texm In this note, a counterexample is given to a theorem published recently by Wolf (1). For convenience we use only random sequences. Let {Xn, n > 1) be a strictly stationary random sequence for which EX12 < cc). If R,(n) = EXjXj+, is the autocorrelation function of (X,) then it is well known parzen (Z), p. 73-75 where R(v) is replaced by R,(n) - (EXl)2] that .** + X,)/n]

limEIC(Xl+ ?&+a,

- EXI I2 = 0

(1)

if, and only if, lim (l/n)

c

n--m

i-a

R2(i)

= (EXI)~.

(2)

In other words, (Xl + . * * + X,)/n converges in mean-square to EX1, if, and only if, Eq. 2 is satisfied by R,. It is easy to see that if limR,(n) n-m

= (EX1)2

(3)

is satisfied then Eq. 2 is true [(Z), p. 751 We ask the following question: If {X,) is a strictly stationary random sequence where EX12 < ~0 and where Eq. 3 is satisfied, will Eq. 1 be true if X, is replaced, for all n, by g(X,) where g(x) is an analytic junction for which E[g(Xl) 1” < co ? The answer, as given by the theorem in Wolf (1)) is yes. However, we now show with an example that the correct answer is, in general, no. Consider the random variables I, Y1, YZ, - - a, .&, Z2, - - - which are independent and where (i) P[1 = l] = p, P[I = 0] = q, p + q = 1; (ii) Yi is, for each i, Gaussian with mean 0 and variance u,2; and (iii) Zi is, for each i, Gaussian with mean 0 and variance uZ2< uU2. Letting X, = IY, + (1 - 1’)2, for all n, it follows that (X,, n 2 1) is a strictly stationary random sequence with EXI = 0 and

R,(n) = puu2+ quz2 = 0,

n=O n > 0.

Thus iimR,(n) n-m

258

= O = (EXI)~

Journal of The Fraddin Institute