that if the second stage law were to hold good down to zero velocity, we should have a certain small residuary resistance (see Fig. 29). This means on the velocity-resistance diagram that the origin for the 1.5 index curve will be situated a short distance up the axis of y; we may conveniently construe this as an approximate parallel to the curve struck from the true origin, when Fig. 32 will represent the manner in which the resistance curve may be supposed to be built up.
§ 55. Some Difficulties of Theory.—In all cases of skin-friction where the index exceeds 1.5 the motion is accompanied by turbulence, and if the value of the index rises to 2, as it would appear to do approximately in the case of the roughened surface, then the dimensional equation (as pointed out by Allen) shows that the resistance is independent of viscosity, and the whole of the energy is expended dynamically in producing fluid motion. Under these circumstances we must regard viscosity as merely acting as a gearing by which rotational motion is imparted to the fluid, although it is difficult to understand how such a gearing can be continually imparting rotation to new masses of fluid without a certain amount of slip; and such slip would betoken an expenditure of energy in viscous motion and necessitate the value of the index being less than 2.
Beyond this it is difficult to conceive of the resistance being independent of the value of viscosity without being independent of the existence of viscosity, which appears to be absurd. Consequently it is probable that, so long as the effect of elasticity of the fluid is not felt, the value of the index, connecting resistance and velocity or resistance and linear dimension, can never reach its limiting value, 2, but must always fall short of it by some small quantity.
It is known, from experiments on resistance in the flight of projectiles, that for velocities approaching the velocity of sound, the index may rise considerably above the limiting value given in the foregoing theory ; and therefore we may expect to find in
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