Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/175

intervals of time. In the elements $h_{3}\ h_{4}$ and $h_{4}\ h_{5}$ the horizontal components of the forces acting on the particle are equal and opposite; therefore the loss of horizontal velocity along $h_{3}\ h_{4}$ is equal to the gain along $h_{4}\ h_{5},$ and the horizontal velocity at $h_{3}$ is equal to that at $h_{5}$. Similarly the horizontal velocities at $h_{2}$ and $h_{6}$are equal, etc., and in general the horizontal velocity component of any particle on one side of $P\ Q$ is equal to that of the similarly situated particle on the other side. But the original state of the fluid is one of no horizontal motion.^[1] This, therefore, is also the final state.

We have consequently shown, in a system such as we have established by the present hypothesis, that the motion imparted to the fluid is eventually given up by the fluid both in respect of its vertical and horizontal components, and consequently there is no continual transmission of energy to the fluid, and no work requires to be done to maintain the motion or to support the plane. The fluid in the vicinity of the aeroplane is in a state of motion, and consequently possesses energy, but under the conditions of hypothesis the quantity is less than any assignable finite magnitude, that is to say, infinitesimal, but the motion remaining in the fluid and the continued energy expenditure are of zero value considered as infinitesimals of the same order. Therefore, adopting a method of expression common in mathematical work (but not so frequently employed in direct physical demonstration), we may say that if we take as hypothesis a small finite load, so that the actual motions of the fluid be small finite quantities, the expenditure of energy in sustaining the load will be zero, neglecting small quantities of the second order.

§ 116. Interpretation of Theory of Aeroplane of Infinite Lateral Extent.—The system of flow deduced in the foregoing article in the case of an aeroplane of infinite lateral extent in an inviscid and incompressible fluid is one that may be classified as a conservative system, the energy of the fluid motion being carried

1. Relatively to the earth.

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