Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/312

mass, for the mean square is always greater than the square of the mean.

The full expression where the velocity is variable throughout the mass ${\mbox{m}}$ is,

${\frac {V}{V+{\frac {\mbox{v}}{2}}+{\frac {\Sigma \ {\mbox{m}}\ (x^{2})}{2\ {\mbox{m}}\ {\mbox{v}}}}}},$

(4)

where $x$ represents the velocity communicated to the particles of fluid in excess or in deficit of the mean, $x$ being accordingly plus or minus. It will be noted that since $x^{2}$ must be always positive the quantity ${\tfrac {\Sigma \ {\mbox{m}}\ (x^{2})}{2\ {\mbox{m}}\ {\mbox{v}}}}$ will be always positive, so that the efficiency will be less than if the mass were handled uniformly. The above expression is of but slight utility from a quantitative standpoint; it is given here as being conducive to exact thought and as being the more complete form of expression (3).

§ 199. Propulsion in its Relation to the Body Propelled.—In the preceding section the subject of propulsion has been treated in the abstract; it has been assumed that the body propelled is far away so that the fluid is unaffected by its presence, and that the fluid as a whole receives momentum.

Now we know from the Principle of No Momentum (§ 6) that, as a whole, the fluid does not receive momentum, and that if it receive momentum in one direction in one part it simultaneously receives equal and opposite momentum in some other part. The result of this is two-fold: (α) we know that the whole of the energy expended in the fluid does not appear as sternward motion, as assumed by Rankine; and (b), the problem becomes complicated by the reaction and motions produced in the fluid by the vessel itself as affecting the conditions under which the propeller is working.

For reasons stated in § 8 it is doubtful whether, under the conditions that ordinarily obtain, the error that arises from

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