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1915]                                THE  PRINCIPLE  OP SIMILITUDE                                    305
or which comes to the same
,,_ic6   _ /av   cv\ a '.   a \ v ' K I '
F, Fl being arbitrary functions of two variables.     And, as we have  seen, F(Q, CV/K) = 1.
For a given fluid CV/K is constant and may be omitted. Dynamical similarity is attained when av is kept constant, so that a complete determination of F, experimentally or otherwise, does not require a variation of both a and v. There is advantage in retaining a constant; for if a varies, geometrical similarity demands that any roughnesses shall be in proportion.
It should not be overlooked that in the above argument, c, K, v are treated as constants, whereas they would really vary with the temperature. The assumption is completely justified only when the temperature difference Q is very small.
Another point calls for attention. The regime ultimately established may* in some cases depend upon the initial condition. Reynolds' observations suggest that with certain values of av/v the simple stratified motion once established may persist; but that the introduction of disturbances exceeding a certain amount may lead to an entirely different (turbulent) regime. Over part of the range F would have double values.
It would be of interest to know what F becomes when av tends to infinity. It seems probable that F too becomes infinite, but perhaps very slowly.]
K. vi.                                                                                                    20locity v. In a uniform regime the heat passing acr.oss is proportional to the time and to the area considered ; but in many cases the uniformity is not absolute and it is necessary to take the mean passage over either a large enough area or a long enough time. On this understanding there is a definite quantity h', representing the passage of heat per unit area and per unit time.