By Shields M. C.

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**Additional info for A Determination of the Ratio of the Specific Heats of Hydrogen at 18 and -190C**

**Example text**

60) This ensemble average is a time dependent function, also for systems in equilibrium. It describes the average evolution of the phase function #, given that at time to < t the value of the phase function f was fo. 3. - > on the left hand-side denotes ensemble averaging with respect to the initial condition fo. This ensemble average is time independent for systems in equilibrium. In an experiment one usually measures an unconditional ensemble average, that is, the system is not prepared in a certain state before the experiment is started.

5c). Integrals ranging over such closed contours can again be written in terms of integrals with respect to x or y as discussed above for the example of fig. 5a. 5). 6). The vector dS is then equal to dx dy (0, 0, 1), and dl points in the anti-clockwise direction. Consider fields F(r) of the form (-u(x, y), v(x, y), 0), with u and v continuous differentiable functions. Since in this case, V x F(r) - (0, 0, Ov/Ox + Ou/Oy), Stokes's integral theorem reduces to, fs dx dy { Ov(x, y) + Ox ~o,~{-dx u(x, y) + dy v(x, y)} .

To appreciate this, consider as an example the Fourier transform of Vx 9F(X), fdX [Vx. F(X)] e x p { - i k . X} = ]" dX Vx" I F ( X ) e x p { - / k . X}] + ik. f dX F ( X ) e x p { - i k . F(k). 4). Since the volume integrals range over the entire space, the surface integral ranges over a spherical surface with a radius that tends to infinity. ) for all X with length R ), I f dS. F ( X ) e x p { - i k . X} i_< <_ maxlxl=R I F ( X ) I J lids. X}! dS - 2 maxlxl=R i r ( X ) l, 22 Chapter 1. the surface integral is zero when the product of the maximum value of I F ( X ) I on spherical surfaces with very large radii R and the surface area of the spherical surface tends to zero as R ~ ~ .