* Where reference is made to “Aerodynamics” it is to be understood as the author's work (published by Constable & Co.).

1 Proc. Inst. Automobile Engineers, Vol. IX, pp. 243 and 253.

2 “Aerodynamics,” §§158, 234 et seq. Also Paper contributed to the San Francisco Congress 1915; Part II, see reprint “The Flving Machine from an Engineering Standpoint,” p. 115, Constable, London. Also Proc. Inst. Automobile Engineers, Vol. IX, pp. 242 et seq.

3 Perhaps somewhat augmented: see “The Flying Machine from an Engineering Standpoint,” Constable, London, p. 118.

4 Compare Fig. 33, Proc. Inst. Automobile Engineers, Vol. IX, p. 247.

5 “Aerodynamics,” §182, et seq.

6 “Aerodvnamics,” Fig. 98, p. 212, Fig. 114, p. 266, also Figs. 117A and 117B, p. 277.

7 “Aerodynamics,” §§184, 187.

8 “Aerodynamics,” Chap. II.

9 “Aerodynamics,” §48. Also Proc. Inst. Automobile Engineers, I.e. ante.

10 “Aerodynamics,” 1907, §§35, 50.

11 For meaning of symbols refer Part IV, §33.

12 See Langley, “Experiments in Aerodynamics,” pp. 9, 23, 25, 64 and 65 (Fig. 11), Smithsonian Institution, Washington.

13 Wm. White, Naval Architecture, p. 441; John Murray, London.

14 Adv. Comm. for Aeronautics, Tech. Report, 1913-14, p. 43. “The Resistance of Spheres.” Also 1915-16, R. & M. 190, pp. 19 et seq. Compare §9.

15 Where the Reynolds number commonly exceeds 10⁸. In the case of a vessel 100ft. in length at 10 knots, the Reynolds number is 10⁸.

16 N_R is the symbol used by the author for the Reynolds number LV/v. Compare §8 and Part IV, §33.

17 For meaning of symbols see §33.

18 “Notes on Model Exoeriments,” Proc. North-East Coast Inst. Engineering and Shipbuilding, Vol. XXXII, 1915. See also Appendix III.

19 R. & M. 1199.

20 R. & M. 1168.

21 §9. Also see Appendix III.

22 Actual determinations made in the wind tunnel also show glaring inconsistencies.

23 Except as concerns the ambiguity in the region of NB=106. See Appendix III.

24 “The Flying Machine from an Engineering Standpoint,” I.e. ante, p. 44.

25 C₀ being chosen appropriate to the Reynolds Number based on L the length of the model. Compare §38.

26 These are minimum values.

27 Had we made use of “augmented” values (see Appendix II) the hull resistance alone would have been 8,250 pounds, and the total (shaft) h.p. =2,800.

28 Augmented value (compare Table, Appendix II).

29 M. Surcouf s figure was 1/30.

30 “The Flying Machine from an Engineering Standpoint,” p. 44, Constable, London.

31 In most of the early Reports and Memoranda of the official Committee the linear parameter chosen has been the cube root of the volume.

32 Compare Appendix II.

33 Proc. Institution of Automobile Engineers, Vol. IX, p. 171, et seq. See also Journal of the Royal Aeronautical Society, No. 190, Vol. XXX, 1926; (Wilbur Wright Lecture).

34 Dimensionally,

ML/T² = (M/T)x(L/T).

35 §§5 et seq. Also §112.

36 Some writers have misrepresented Newton as having put forward this “law” as applying to air. Langley, for example, “Experiments in Aerodynamics,” p. 24; p. 89 line 2; p. 105 last 3 lines. Compare author's “Aerodynamics,” §238, p. 366.

37 First enunciated by Vince, Phil. Trans. 1798.

38 Compare “Aerodynamics,” §§149 et seq. It is well here to call attention to the differ-- ence between the theory of the Newtonian medium and the Newtonian principle as applied by Rankine and by the present writer.

39 Actually the law holds for angles considerably greater than included in this definition.

40 So far as the present discussion has carried us.

41 A'/a varies as a function of 11 the aspect ratio.

42 “Aerodynamics,” §§112 et seq.

43 Compare Part V.

44 See footnote, §33.

45 The definition is given in the concluding paragraph of §210. “Now, we may evidently regard the aerofoil, with its accompanying peripteral system, as the equivalent of the hypothetical device which we have temporarily assumed. The peripteral system actually constitutes a kind of tool or appliance by which the aerofoil is able to deal in effect with more air than actually comes within its sweep. This extended ' sphere of influence ' of the aerofoil will be termed the peripteral zone, and its cross-section, {(1+є)/(1−є)}xa is the peripteral area.”

46 Compare §28 herein.

47 “Experiments in Aerodynamics,” 1891 (Smithsonian Institution).

48 “Experiments in Aerodynamics,” see ref. §6, footnote.

49 It has already been demonstrated that all direct resistance, including both skin-friction and that due to errors of form, may for practical purposes be included as varying directly as V².

50 “Aerodynamics,” Prop. I, §164.

51 “Aerodynamics,” Prop. II, §164.

52 “Aerodynamics,” Prop. Ill, §164.

53 “Aerodynamics,” p. 254.

54 “Aerodynamics,” Prop. IV, §165.

55 Compare Appendix IV.

56 “Aerodynamics,” Prop. V, §166.

57 A high aspect ratio is beneficial because it results in a diminished direct wing resistance. The larger size gives relief on account of the scale effect, i.e., a lower Reynolds number, and as a consequence a somewhat lower value of C₀.

58 See Appendix II. Also “The Flying Machine from an Engineering Standpoint,” p. 94; Constable, London.

59 The difficulties arising in the application of the Newtonian principle have already been pointed out and references made to the discussion in §§5 to 10 of the author's “Aerodynamics” (l.c. ante).

60 Published in the Journal of the Royal Aeronautical Society, October, 1926.

61 With a superposed motion of translation.

62 As disclosed in a paper read in Birmingham in 1894. See “Aerodynamics,” §§133 et seq.

63 “Aerodynamics,” Chapters III and IV.

64 §27.

65 “Aerodynamics,” §83, p. 110, concluding paragraph.

66 See Part II, §21.

67 If the case be that of gliding flight the imaginary planes may be vertical when the vortex reaction will be in direct opposition to gravity. A component of gravity then supplies the propulsive force. The objection is thus shown to be frivolous.

68 The values of e and K are as given in “Aerodynamics,” §181. A similar Table is given in “Aerodynamics,” p. 312.

69 The elliptical plan-form of aerofoil is here assumed, as adopted by the author as a standard.

70 Yet it is surprising how little is the difference between the crude rectangular plan-form of uniform section and the more modern tapered wing with terminal “wash-out”; it would seem that once the general principles are mastered, or appreciated, it is by no means easy to design, unless by deliberate intention, a really ineffective form of aerofoil.

71 In other words greater than the critical value, at which according to report instability is to be feared. For most purposes it is sufficient to make the distinction as above and below 10⁶.

72 The author has found some difficulty in agreeing the figures. See Appendix I.

73 As far as possible well recognised symbols or descriptive symbols have been adopted.

74 In some of his earlier writings the author used a “double surface coefficient” of skin-friction designated by the symbol ξ which betokened the resistance of a flat plate in edgewise presentation in terms of the same plate normal to the wind. Thus,

0.6 ξ = 2C₀ or C₀ = 0.3 ξ.

At low values of N_R the values of ξ tabulated by the author in his “Aerodynamics” ranged from .01 to .03, corresponding respectively to C₀ .003 and .009. (See chart. Fig. 2).

75 Numbers of Tables as referring to R. and M. 1627 relate to position on page and are given in ordinary numerals. Numerical references to Tables in text are given in Roman numerals.

76 In Tables V, VI and VII the higher values of K_L are entered first, in order to correspond Veith plottings,

77 If taken from Fig. (Appendix II) on the basis of 31,000lbs. N_E=1.55xl0⁷.

78 In all probability too low: we are wearing the “rose-coloured spectacles.”

79 The difference, less than 1 per cent., is in part due to the use of the slide-rule and in part to the figures not having beep taken beyond the third significant figure.

80 The value of C₀ = 0.0013 assumed is higher than that taken for the aerofoil owing to the lower value of N.

81 See Appendix IV and V.

82 Example: Chord = 4ft.; V = 54ft. sec. Then N = (4 x54)/.000154 = 1.4 x 10⁶

83 §31,

84 R. & M. 110, 1913-14, pp. 94, 99 and 100. N_R = 1.3 to 1.6x10⁴.

85 Before the term “Reynolds number” came into use, which was about the year 1924, the basis of comparison was often expressed as LV. It being legitimate to regard the kinematic viscosity of air as constant, this really amounted to the same thing as making use of the Reynolds number. Actually the author was the first to deduce and apply thel expression LV/v in connection with the dynamics of flight (see “Aerodynamics,” §38).

86 See footnote §33; 0.6 ξ = 2C₀ or C₀ = 0.3 ξ.

87 N_R=1.4xl0⁶.

88 As stated. See Programme, R.Ae.S. Garden Party, 10th May, 1936. These figures presumably include body resistance. On the other hand the aspect ratio is often greater than 12.

89 It is not easy to determine the normal flight speed of a bird of any particular species. The fact that the Reynolds number forms a connecting link between weight and velocity is sometimes a great help, for the weight and the wing, plan-form and aspect ratio can be ascertained from the dead bird as shot, and the normal flight velocity inferred with a reasonable degree of certainty.

90 A notable exception is that of the French writer Mouillard.

91 To judge from the books written by ornithologists it might be imagined that the writers were unaware that birds fly, no mention is made of the data relating to the function of flight. The weight of the adult bird is rarely given, almost by accident if at all. The normal flight velocity, if stated (which is rare) is usually wrong, and often absurd. The sail plan is never given, for that one has to go to Mouillard. No observations are forthcoming on the important subject of gliding angle, and so on, the list is by no means exhausted. But the pretty colours of the plumage are described in detail, also some of the seeds that are found in its “little inside,” and the descriptions are written in an imitation of literary style, the writers studiously avoiding anything in the way of tabular form or logical order. Each article has to be read from end to end to make sure that the information wished for is not there (it usually is not). In brief, the typical ornithologist seems to combine in one personality a power of observation equal to that of an immature child with a varied capacity of expression worthy of a professional journalist. The entomologist dealing with insects that fly is often not much better.

92 The larger models weighed 1¼ to l½Ibs. “Aerial Flight,” Vol. II, Figs. 15, 16 and 17. These had for their primary purpose to demonstrate the principles of stability. Such models with their relatively high speed of flight are not easily made the subject of exact aerodynamic measurement. Even then the value of N_R was considerably less than 10⁵.

93 This would affect the weight relation, but not that derived from measurement.

94 One of the original aerofoils is now preserved in the Science Collection, South Kensington Museum.

95 Advisory Committee, Technical Reports, 1913-14, p. 87.

96 It might be thought that when W is a variable the value of N_E taken on a basis of weight {Appendix II) must vary also, but this is not so, the weight basis relates to the condition of least resistance.

97 Fig. 21, Appendix II.

98 See Technical Reports, Advisory Committee for Aeronautics, 1915-16, pp. 21, 22.

99 Proc. Royal Institution of G.B., Vol. 28, Part IV, No. 135. Discourse by Professor Melvill Jones; pp. 648 et seq.

100 R. & M. 110, 1913-14, p. 100; Table 12.

101 This Table is illuminating: the lower efficiency at lesser values of K_L and the higher efficiency at greater values, suggests that the sectional form is better adapted to heavy loading than in the case of the author's 1894 model.

102 Table 4, p. 94.

103 R. & M. 607.

104 All we are doing is to invert the calculation by which the figure 203.5 was initially derived.

103 The linear parameter 0..666 is the chord dimension of the model aerofoil in feet.

106 ξ is of the dimensions L/T² (acceleration).

107 The weigh-bridge or other instrument used in weighing is a mass comparator.

108 For justification, see “The Flying Machine from an Engineering Standpoint,” pp. 96 et seq.

109 §8.

110 As in chart (Fig. 2) plus 25 per cent. Compare “The Flying Machine from an Engineering Standpoint,” p. 96 et seq.

111 “Aerodynamics,” §§33, 34, 35. Also §40 as concerns energy relation.

112 Without a definite abandonment of the theory. It is always possible, when a quantity y is normally a single valued function of another x, that for some range of values y may be a multi-valued function, and,j as a limiting case, the value of y as a function of x (or vice versa) may be indeterminate. There is another aspect to be considered; as the value of N_R, changes and the system of flow undergoes a modification, a time factor may enter into the problem, so that the relations (in the present case) of C₀ to N_R , even if quite definite in the steady state, may depend, collaterally, on the rate of change, that is upon dN_R /dt. Then there is the possibility of both these influences being simultaneously at work, namely, an unstable or quasi stable condition complicated by the rate of change.

113 R. & M. 1199, Aeronautical Research Committee, December, 1928.

114 R. & M. 190 (1916). Here the instability signalising change of system of flow is given as taking place at a value of 2x(VI/v), but what this cryptic expression means the author cannot say.

115 §21, also see “Aerodynamics,” Chapter VII.

116 The author referred to this as a “forced wave” in his Patent Specification 3608 of 1897.

Note on page 115 * Trans. I.N.A. 1936.

117 Vol. XXX, No. 190, p. 603.

118 R. & M. 73, 1912/13.

119 “Aerodynamics,” §126, Fig. 83.

120 Perhaps aided by a catapult.

121 This is suggested and illustrated in my paper on “The Aerofoil,” i.e., note Fig. 14. See also reprint “Two Papers,” p. 25.

122 I think it is quite proper to refer to this theory as the Lanchester-Prandtl Theory. I do not grudge Prandtl his share.

123 The membership of this Board includes representatives from Canada, England, France, Germany, Holland, Italy and Japan, designated by a leading aeronautical organisation in each country, in addition to eight members of the Trust Corporation controlling the Daniel Guggenheim Fund for the promotion of Aeronautics.

124 “The Xheory of Dimensions and its Application for Engineers.” Lanchester, Pub. Crosby Lock wood & Son, London.

125 Even earlier. My first pendulum accelerometer was made by my own hands in 1889.

126 This is a phenomenon that should be more readily observed in the case of aerofoils with square or bluff extremities, such as in the case of one of rectangular plan-form, especially when of low aspect ratio. In effect it would represent aerodynamically a slight extension of the span and perhaps justify the rather greater peripteral area given for low values of n in the table quoted from my “Aerodynamics” than due to the formula пS₂/4.

127 Congrave text of paper, §38.

128 They are thus assumed to differ only in size.

129 This supposition is precisely equivalent to regarding the aerofoil of infinite span carrying a correspondingly infinite load, but gives a great degree of realism. In both connections the motion engendered is two-dimensional.

130 Vol. IX, pp. 223, et seq.

131 It is no use weighing “sugar-plums” on a chemical balance, or employing a micrometer to gauge acorns.