Page:Aerial Flight - Volume 2 - Aerodonetics - Frederick Lanchester - 1908.djvu/88



§ 38. Permanence of Stability.—The theoretical investigations of the preceding two chapters constitute the proof, under the restricted conditions of hypothesis, of the longitudinal stability of an aerodone in flight, the whole demonstration being a quantitative analytical version of the theory of stability enunciated in §§ 3, 4 (Chap. I.).

The permanence of stability, according to the present proof, rests definitely on the fact that the phugoid curve consists of a succession of phases repeated, without change of form, an indefinite number of times. If the substitution of real conditions for those of hypothesis be found to result in any progressive change in the form of the curve, that is, if the constant C be found to undergo a change (for a given value of Hn), or more broadly, if the constant K suffer any progressive change of value, then the question of permanence of stability is greatly complicated, and further investigation will be required.

There is one particular case of the phugoid curve, even under the supposed conditions, in which the stability must be regarded as indeterminate; this is the semicircle when C = zero. The fourth condition of hypothesis, § 19, is " that the size of the aerodone is small in proportion to the minimum radius of curvature of its flight path." Now, for the purposes of the theoretical investigation, this condition may be supposed fulfilled by the aerodone being assumed to have no size at all, but in the case of the semicircular flight path this is not sufficient, for the "cusp" must be considered to be a portion of the flight path {[c|66}}