where R varies as V will depend, as already stated, upon the size of the containing vessel.
§ 53. Change in Index Value (continued).—We have so far confined ourselves to the discussion of the first change of index, that which takes place when the curve passes from the Stokes to the Allen stage.
The second change of index value evidently takes place when the motion of the fluid becomes turbulent, for it is then that the conditions leading to the normal law of skin friction are violated, and the energy relation becomes disturbed. In all probability also the V2 law, in cases involving other than pure skin-friction, is closely associated with the phenomenon of discontinuity. A system of flow of the discontinuous type is almost certainly accompanied by resistance following the V2 law.
The conclusions of this and § 52 at present lack experimental demonstration. There would appear to be some evidence to show that the Stokes stage may exist independently of the size of the vessel; this at least is a conclusion reached by Allen. If this should prove to be the case the explanation here given will need modification.
§ 54. The Transition Stages of the Characteristic Curve.—The junction or transition portions of the curve connecting the various stages are not angular as shown diagrammatically in Fig. 31, but pass gradually from the one to the other. The transition stages, however, are not such as to mask the distinct individuality of each portion of the curve, but merely enough to render uncertain the precise point at which the change of "law" takes place.
It appears that there is a small departure from the exact expression given, both in the second stage (as found by Mr. Allen) and in the third stage. In the latter case we may suppose that when the velocity becomes very great the geometrical form of the lines of flow becomes sensibly constant, and such resistance{[c|71}}
