## ABSTRACT

General formulae are derived giving the lift, thrust and power when the wing motion is specified. The formulae are applied to twenty-five insects for which quantitative data are available. Average values for lift and drag coefficients,

*C*_{L}and*C*_{D}, are derived by equating the weight to the vertical force and the thrust to the horizontal drag of the body.The large drag and lift coefficients obtained for insect flight are attributed to acceleration effects. There is a distinct correlation between and the ratio of the flapping velocity of the wings to the linear velocity of flight. When this ratio and therefore the accelerations are small, the force coefficients do not exceed those to be expected for flat plates. Owing to the nature of the assumptions and approximations made, the values derived for

*C*_{D},*C*_{L}and*C*_{D}*/C*_{L}are minimum values.Other characteristics of insect flight are discussed. In general, insects fly in such a way as to minimize the mechanical power required. In most, but not all cases, the useful force is the one perpendicular rather than parallel to the relative wind. The wing tips should move in a figure 8, the down beat should be slower than the up beat, and the majority of the necessary force must be supplied on the down beat.

Figures are given using the data from the twenty-five insects considered, showing average relations between power, specific power, mass, acceleration forces, force coefficients and geometrical dimensions. The power per gram, the ‘wasted power’, and the force coefficients all increase as the importance of the acceleration forces increases.

When plotted as functions of mass, quantities involving the power show much less dispersion than quantities involving the geometrical dimensions. This is taken to mean that despite the diversity of insect form, as ‘power plants’, they are all essentially similar.

A table of the observed or adopted flight parameters (frequency of beating, mass, wing area, velocity of flight, amplitude and orientation of wing motion) is appended.

## INTRODUCTION

The problem of natural flight is one of long-standing interest, both intrinsically and in connexion with mechanical flight. Rayleigh (1883) was perhaps the first to give a satisfactory explanation of the soaring flight of birds, and Walker (1927) gave a satisfactory quantitative discussion of the flapping flight of birds in a particular case. Hoff (1919), using the data of Demoll (1918), attempted to bring the flight of insects into the domain of conventional aerodynamics by showing that lift coefficients of reasonable value were required. He used only the linear velocity of flight to evaluate these coefficients and ignored the motion of the wings. He also pointed out the similarity between the flow pattern around an insect and around a propeller or actuator disk, a similarity which must hold from momentum considerations, independently of the action of the wings (Bairstow, 1939). Demoll (1919) indicated several errors and omissions in Hoff’s work, so that in a number of cases the lift coefficients required were inordinately high. Finally, a most thorough experimental investigation was carried out by Magnan (1934), who was, however, unable to explain his results theoretically.

The difficulty of high lift coefficients has not been removed by other investigations which have indicated lift coefficients larger than those normally expected in windtunnel measurements. It has, therefore, been believed that insects must utilize some special mechanism in flight not present in conventional aerodynamic phenomena.

In this paper the mechanism of insect flight will be examined in detail in order to find some possible explanations for the high lift coefficients, as well as for other characteristic features of insect flight. General expressions will be obtained for the force acting on a wing surface element moving in an arbitrary manner. This force will be resolved into lift and drag components with respect to the relative wind, and integrated over the surface of the wing and averaged over the period of a wing beat in order to determine the total average vertical and horizontal forces exerted by the insect. The former is the required vertical force (weight of insect), the latter the required thrust (drag of insect’s body). Average values of the lift and drag coefficients of the wings, and , can then be determined, since there are two known forces given, and two unknown coefficients to be determined.

The scalar product of the vector force on the wing element into the vector velocity of the element with respect to the insect, integrated and averaged as before, gives the mechanical power exerted by the insect in flight.

The absolute magnitude of the total force on the wing element, integrated and averaged, will give a lower bound to the force coefficient for the average total force, .

The power, the lift, drag and total force coefficients computed for the twenty-five insects for which data is available, will, when plotted against the other parameters of insect flight such as mass and ratio of flapping to linear velocity, reveal a number of interesting and systematic characteristics of insect flight. The flight data, summarized in Table i (p. ), were taken from the work of Magnan, and were supplemented by measurements on specimens of the U.S. National Museum.

For those who wish to draw conclusions from this work without verifying the mathematical details, it may be said that the principal mathematical problem is to justify replacing the instantaneous velocities by suitably chosen averages over position and time. Once this process is admitted practically all of the qualitative conclusions may be reached by drawing vector diagrams similar to those of Fig. 4, with these average velocities for any particular case. In fact, if the averaging precepts are admitted, quantitative conclusions can be reached by drawing to scale, and with all small components such as *v* and *w* included, diagrams like Fig. 4.

## DERIVATION OF THE FUNDAMENTAL FORMULAE

**dF**

_{s}on an element of wing surface

*c*(

*r*)

*dr*is given by and the drag force is given by The quantities which appear in these two formulae are defined as follows (see Fig. 1). Vectors are denoted by bold face, their scalar magnitude by two vertical lines.

**1, m, n**are mutually perpendicular unit vectors parallel to the axis of the wing, perpendicular to the axis of the wing in the instantaneous plane of beating, and perpendicular to this plane, respectively. The plane of beating is not the plane of the wing, but the plane in which its axis is moving,

*p*is the air density.

*W*is the relative wind or velocity of the surface element of the (right) wing with respect to the air, and is given by Ω is the instantaneous angular velocity of the wing element, and U = (0, – (

*V*+

*v*),

*— w*) is the velocity of the air with respect to the insect’s body.

*V*is the velocity of flight and

*v, W*are the induced velocities.

*c*(

*r*) is the chord,

*r*and

*t*are the independent variables – distance along the wing and time. In the general case all of the quantities appearing in eqs. (1) and (2) (except

*ρ*and

*c*(

*r*)) are functions of

*r*and

*t*. If the wing is assumed not to twist or bend, the unit vectors and Ω will be independent of

*r. C*

_{D}evidently represents the profile drag coefficient, since the relative wind includes the induced velocities.

*v, w*, or velocity increments of the slipstream, are given from momentum theory (Durand, 1935). They are independent of the mechanism of the wing action, and are obtained in first approximation by requiring that

*v*and

*w*at the insect have one-half the final value necessary to provide the necessary vertical force and horizontal thrust:

*R*is the length of the wing,

*M*is the mass of the insect,

*g*is gravity, and

*S*

_{b}the body cross-section area and

*C*

_{Db}the body drag coefficient. Since, by division of the above,

*v/w=T/L*, and for the insects considered the thrust

*T*is small compared to the vertical force

*L. v*is small compared to

*w*, and will be neglected, since

*w*turns out to be small compared to the other velocities involved. The solution for

*w*is given by Eqs. (1) and (2) express the assumption that the determining velocity (squared) for both lift and drag is the component of the relative wind perpendicular to the axis of the wing, which is |

**1**×

**W**| = |

**W**–

**W. ll**|. The subscripts

*s*and

*p*indicate perpendicular (senkrecht) and parallel

*to that component*,

**W –W.ll**. The last factor in braces gives the direction cosines of the forces. The drag is assumed to be perpendicular to the axis of the wing, hence the subtraction of the

**1**component of

**W, W. ll**in the direction cosine. Forces along the length, or axis, of the wing tend to cancel for lift and thrust on the upper and lower half of a wing beat and overall between the right and left wing, since they are in opposite directions for opposite wings. Also they do not contribute to the power (eq. (8)) since they are perpendicular to the displacement of the wing elements, referred to the insect.

The assumption that the force is proportional to the square of the relative wind component perpendicular to the axis of the wing is the conventional one in helicopter and sideslip performance calculations, and applies primarily to steady flow. Steady flow is far from realized in flapping flight, unless the variable or flapping components of velocity are small compared to the linear velocities, or the flapping velocity changes by a small fraction of itself over a displacement equal to the chord. The assumption of the steady state force formula is made primarily so that the computed results for *C*_{L} and *C*_{D} can be compared with conventional aerodynamic data, as will become apparent from the subsequent discussion. However, there is indirect evidence given below that this assumption is justified, at least in part, for the problem under consideration.

*yz*plane, as in Fig.

*1*, the unit vectors 1,

**m, n**for the right wing are, in the

*x, y, z*system, If

*θ*, the inclination to the vertical of the normal to the instantaneous plane of beating (Fig. 1), is a constant, i.e. the wings beat in a plane, then

*Ω = dϕ/dt*. With these two specializations, eqs. (1) and (2) give (bars indicate average) The range of integration over

*r*is the length of the wing; o to

*R*, and in

*t*over the period 1/

*v*of a wing beat,

*v*being the frequency of flapping.

^{*}is from eq. (8) Here the first term in the braces is the exact expression for

*P*

_{Fa}. The expression for power on the basis of the momentum theory (using eqs. (4) and (5)), is

*VT+wL*, which is the rate of change in kinetic energy in the slipstream on passing the insect required in order to provide the net thrust and lift

*L=Mg*. Thus when

*C*

_{D}=

*0*(‘ideal’ fluid) the power determined from momentum theory and the power determined by integrating force × velocity (i.e. first term in { } of (13)) for each element of the wing agree exactly, as they should.

This is indirect evidence that the assumed formula for F_{s} (eq. (1)) gives a correct result when integrated in a case of periodic motion. The agreement between the formulae for the power computed by the two different methods corresponds, in the case of a simple lifting airfoil of finite span, to the agreement between the work done against induced drag, and the increase in kinetic energy of the trailing vortex sheet.

In practice errors would be introduced by the fact that *v* and *w* are not constant in space and time, but the assumption of their constancy is a standard one in aerodynamic practice, and the error committed is not large.

*θ*, C_{L},

*C*

_{D}are assumed to be constant with respect to

*r*and

*t*, and to have the values If

*P*is the period of a wing beat, the subscripts 1 and 2 refer to the time intervals o <

*t*<π/ ω

_{1}(down beat) and π/ω

_{1}

*<t<P*(up beat), respectively.

*A*is the amplitude of motion in the plane of beating, and η = (

*a*– 1)/(

*a*+1), where

*a*is the ratio of duration of down beat to up beat.

*η*and δ

*θ*are assumed small, but

*ξ, ζ*are not so restricted. The double sign for takes account of the possibility that leading and trailing edge of the wing, or pressure and suction side, may be interchanged on the up and down beat. This does not affect the sign of .

The principal difficulty at this point is to obtain a satisfactory approximate expression for the radical ; once this is obtained the integration is straight forward. No single approximation could be found which was satisfactory for all values of the ratio of the flapping to linear velocity *(rdϕ/dt)/V* and the time. This ratio varies from zero to much greater than unity over the length of the wing and period of a wing beat, and becomes both positive and negative.

One method is to replace the variable terms in eqs. (10), (11) and (13) by averages. Two preliminary remarks are first necessary :

*a*, and

*a*to

*P*, then the average value for the entire interval is gives by In what follows,

*P*corresponds to the period of a wing beat and the two intervals to the period of the down and up beat, π/ω

_{1}and π/ω

_{2}.

*c*(

*r*) is a simple function, o at the end of an interval o to

*R*, and positive in between, like a semi-elipse or a rectangle. We wish to integrate an expression of the form The

*a’s*are assumed such that the expression converges. Now if 5 is the area ,

*I*can be written Now for a simple geometrical figure as above , etc., are all not so very different and one could approximate the desired integral by substituting a single average value for

*r*for all those appearing, that is The best choice would be the one for which the corresponding

*a*was the largest. In the above case presumably

*a*

_{2}was the dominant coefficient.

*(rdϕ/dt)IV*was (1) small, and (2) large, and the resulting expressions could then be integrated. In this way expressions corresponding to (16) above were obtained in which the coefficients

*a*

_{0},

*a*

_{1},

*a*

_{2},

*a*

_{3},… were constant or easily integrated trigonometrical functions of time. Average values were fixed for r( in ( 10) and ( 11 ), in ( 13 )) which fitted both approximations moderately well. Also averages for cos

*ϕ*, sin ϕ, cos

*ωt*were chosen which fitted both approximations moderately well. (Both expansions (1) and (2) above are still fairly good when

*(rd*ω/

*dt)/V*∼ 1.) These selected averages were then substituted directly into the original eqs. (10), (11) and (13). As will be seen by comparing the equations below with eqs. (10), (11) and (13), this process simply amounts to replacing all the terms in eqs. (10), (11) and (13) by shrewdly chosen constant values and integrating in two steps. When this is done the results are as given below.

*S*is the total area of both wings. Let the average velocity components in parentheses in eqs. (10) be indicated by and similarly for

*W*

_{px},

*W*

_{pv},

*W*

_{pz}

*in*eq. (11).

*r*, were determined by numerical integration over the wing form of each insect. Eq. (10) gives Eq. (11) gives The quantities appearing in these two equations are defined as follows :

*J*

_{0}

*(A)*and

*J*

_{1}

*(A)*are Bessel’s function of the first kind obtained from the expansion of cos

*ϕ*, sin ϕ, where ϕ

*=A*sin

*ωt*.

*A, B, C*and

*D*one imposes the condition that the sum of the horizontal forces provide the required thrust and that the total vertical forces provides the lift

*L = Mg*, then Then by Cramer’s rule, This gives, using the substitution indicated for

*A, B, C, D*, the expression for the average lift and drag coefficients and in terms of the observed flight parameter for any given insect.

arid in (24) are computed from the eqs. (23) and (19).

*W*

_{m}and

*W*

_{n}, times a correction factor

*k*which was a very slowly varying function of the ratio

*W*

_{m}

*/W*

_{n}, taken as constant. Thus From that step on the resulting integrations could be carried out almost exactly. The resulting formulae are exceedingly long, since a different set of formulae is required for each of the different sign possibilities for

*W*

_{m},

*W*

_{n}and

*C*

_{L}on the up and down beat.

This second method has the advantage of showing analytically and directly the effect on the vertical force, thrust and power, of varying the flight parameters *C*_{L}, *C*_{D}, * θ* and ω on the up and down beat. The effect of independent variations appear as linear terms in

*ξ, ζ*, η and δ

*θ*, while the interactions appear as cross-products of

*ξ*

*ζ*, η, and δ

*.*

*θ*Check computations by the two methods gave results in fair agreement. All of the numerical results given in this paper were obtained by the first method.

## DETERMINATION OF FORCE COEFFICIENTS

Applications of eq. (23) to determine *C*_{L} and *C*_{D} for the twenty-five insects given in Table 1 were made under two assumptions: *ξ* = *ζ* = 0, or equal force coefficients on the up and down beat, and *ξ* = *ζ* = 1, or zero force coefficients on the up beat. It was felt that these represented the two limiting cases between which the actual practice must lie. The results are shown in Figs. 2 and 3, which give the average force coefficients for the entire wing beat or the value on the down beat, respectively. The length of the horizontal line represents the range of solution as the drag coefficient of the body varies between zero and unity, the points being plotted for zero-body drag coefficient.

It will be observed that for the case of zero force coefficients on the up beat the ratio of lift to drag does not exceed 3 or 4 to 1, which is quite acceptable aerodynamically. Hence large lift coefficients are always associated with large drag coefficients. All coefficients are positive, the largest lift and drag coefficients being 5 ·4 and 2 ·6, respectively. In the case of equal-force coefficients on the up and down beat, there are two instances of negative lift and drag coefficients, and six with zero or negative drag coefficient. The largest lift and drag coefficients are 11 and 20, respectively.

Under both assumptions, therefore, somewhat larger force coefficients are derived than are met with in conventional aerodynamics, but the assumption of equal-force coefficients on up and down beat leads to the additional anomalies of zero or negative drag and lift coefficients. It is, therefore, concluded that the assumption of zero force coefficients on the up beat must represent a much closer approximation to the truth than equal force coefficients on the up and down beat. In other words, the insect derives the great majority of the useful flying force on the down beat (Guidi, 1938; Holst & Kuckemann, 1942). This conclusion is reinforced by comparing points for the same insect on Figs. 2 and 3. The required lift coefficient for forces acting on the down beat alone is in most cases only slightly greater instead of twice a great, as might be expected, than when force acts on the wing during the entire beating cycle. In a few cases, it is less. Except for the ‘anomalous cases’, the drag coefficient is always a great deal less for the down beat alone than for the entire cycle.

There is a simple explanation for the conclusion that the great majority of the useful flying force must come on the down beat. One draws the components of the average (with respect to *r* and *t)* relative wind in the *yz* plane (Fig. 4). It is specified that the mean flapping velocity is the same order of magnitude as the velocity of flight *V*, say, within a factor of 3 larger or smaller. This range covers most of the insects considered. Fig. 4 illustrates a number of such diagrams showing various choices (1/3, 1/1, 3/1) for the ratio of flapping velocity to the velocity *Ū, V*_{F}*/ Ū*, and the inclination * θ* of the plane of beating. Fig. 4 is essentially a geometrical interpretation of eqs. (19) and (20). In this way one can determine to within a quadrant, referred to axes parallel and perpendicular to the relative wind, the direction in which the resultant force

**F**

_{s}

**+ F**

_{p}on the wing during the up or down beat must lie. Now the average resultant force for both up and down beat

*must*provide the necessary vertical force

*Mg*and thrust. This is a force which is predominantly up, with a small forward component. If it turns out, as is usually the case, that no matter how

*θ*, the inclination of the plane of beating is chosen, that for the up beat

*any*force within the 90 ° uncertainty has small or negative components in the direction in which the average resultant

*must*lie, then

*any*force acting during the up beat is undesirable. The insect will therefore tilt his wings to achieve as nearly zero force coefficients on the up beat as possible.

If the flapping velocity is larger than the linear velocity and *θ* is ∼30 ° or less (Fig. 4 e), there is some advantage for the insect to reverse the direction of the lift (*C*_{L} force) on the up beat, by interchanging either the leading and trailing edge on the up and down beat or the pressure and suction side. In the calculations this assumption was made in those cases in Fig. 2 for which *θ* was taken to be 30 °— all except the six on the *C*_{D}*= θ* axis. There is some evidence that this ‘reversed flapping’ actually occurs, judging from the drawings and photographs in the literature. For ‘reversed flapping’ calculations the negative in the double sign in eqs. (15) and (19) is chosen.

When reversed flapping takes place there is also some advantage to the insect to move his wings in a tilted figure 8, with the steep side on the down beat (δ *θ* > 0). In this way the average resultant forces on the up and down beat more nearly approach parallelism, and the resultant on the up beat is more nearly in the desired direction. Observation is in agreement with this (Magnan, 1934; Schröder, 1928).

Fig. 4 also serves to show that the force perpendicular to the relative wind or *C*_{L} force is the one which most commonly provides the useful or flight-enabling force. Observations indicate (Guidi, 1939; Kukentahl & Krumbach, 1927 –34, pp. 573 –5; Magnan, 1935, pp. 61, 97,104,112, 114;Marey, 1890) that *θ* is usually 90 ° or slightly less when the ratio is small and can decrease as this ratio increases. Drawing the relative wind diagrams, as in Fig. 4, will show that the *C*_{L} force is more nearly in the direction in which the average resultant must lie ; the *C*_{D} force is at best usually at right angles to the necessary resultant. For an exception to this statement, see *d*, Fig. 4, down beat. Here both *C*_{L} and *C*_{D} forces are nearly equal and equally useful. Thus the insect, in common with all successful flying machines, primarily depends on the force at right angles to the relative wind for flight. Magnan (1934) sketches one exceptional insect, the *Syrphus* fly, for which the *C*_{D} force is the useful one. Use of the *C*_{D} force is one way to obtain very large force coefficients at the price of considerable power. Other examples of the use of the *C*_{D} force will undoubtedly be found when more experimental data is obtained. They will occur when *θ* is near 90° when hovering, or greater than 90° in forward flight. In any case the significant force, perpendicular or parallel to the relative wind, can always be determined for any given mode of flight by drawing the relative wind diagram as in Fig. 4.

There remains to explain the occasional large lift and drag coefficients which appear in Figs. 2 and 3. To this end reference is made to Figs. 5 and 6 in which the minimum total force coefficient, determined from eq. (25), is plotted against the ratio of the root-mean-square (with respect to *r* and *t*) flapping velocity to the velocity of flight. This ratio measures the importance of the acceleration or inertia forces ; when small the inertia forces are negligible. It will be observed that there is a distinct correlation between this ratio and the total force coefficient; when this ratio is small the force coefficient does not exceed those values normally expected in aerodynamic practice.

Thus the abnormally large lift coefficients derived by previous investigators must be attributed to acceleration effects. Both theoretically and practically there is no upper limit to the force coefficient referred to the velocity squared if acceleration forces are permitted. Thus for an impulsive start from rest, as in the case of an oar which is jerked, the terminal velocity may be very small and the force large. Hence the force coefficient referred to the square of the average velocity will be very large. This conclusion holds for both potential and non-potential flow, though in the former case the motion must not be perfectly cyclic.

Part of the high lift coefficient may be due to a slotted wing or flap effect. For those insects with two pairs of wings which overlap, the anterior always overlay the posterior, which is the correct arrangement for a slotted wing. In some cases the posterior is hinged by hooks to the anterior, in which case the posterior may act as a flap.

In Figs. 5 and 6 the vertical lines run up to the values of , as determined from the separately derived values of and given in Fig. 3. Computations of , were also made for the case of equal force coefficients on the up and down beat (*ξ* = *ζ* = 0) and for assumed *θ* = 90°and o° (not given here), and all led to essentially the conclusions above. In the case of the remaining figures in this paper, they all refer to zero-force coefficients on the up beat *ξ*= *ζ*= 1). Calculations were also made for *ξ* = *ζ*= 0 but are not given, since they indicated in some cases too small or negative values for the power, corresponding to the zero or negative values of *C*_{D}.

Since the observations did not provide values of *θ* they had to be assumed, though in some cases they could be inferred from the drawings of Magnan. This assumed value influenced the value of the double amplitude *2A of the* wing motion in the plane of beating, since the double amplitude projected on the vertical (*xz*) plane, 2ψ, was the observed quantity given by Magnan. Hence the smaller the assumed *θ*, the larger the derived *2A*, which would vary between its projected value and 180° as *θ* varied from 90° to 0°. The effect of the assumed *θ* on the derived minimum total force coefficient is shown in Fig. 7, where it can be observed by tracing out a curve for a single insect that the force coefficient can at most increase by a factor of 4 or 5 to 1 as *θ* varies from 0 to 90°. The variation is much less for those insects with a projected double amplitude close to 180° and for those insects for which no projected amplitude was given, so that a constant amplitude in the plane of beating had to be assumed. The range of solution for 0 < *C*_{Db} < 1 is not given in Fig. 7 ; the range was negligible. Assumed values of *θ* and *A* in Table 1 are in parentheses.

The actual values of *θ* as used for the computations of *C*_{L}, *C*_{D} (Figs. 2 and 3) and the remaining figures involving the power were chosen as indicated in Table 1. In general, for , *θ* was assumed to be 30°, whereas if this ratio were much less than 1, *θ* = 60° was assumed. The values of 0 chosen by this rule were the least values compatible with the available drawings of Magnan and others. The consequence of such a choice is that *A* is a little too large, and the derived values of and and the power a little too small. This is considered to be ‘advantageous’ to the insect in that the necessary power is a minimum. The range of solution as *θ* is varied can be estimated from Fig. 6.

From another standpoint also the derived average values of and are minimum values. Since the actual instantaneous values of these quantities must vary with *r* and *t*, at some time the instantaneous values must exceed their averages.

According to Table 1, col. 8, it will be observed that the down beat is slower than the up beat, or ω_{1}<ω_{2} (see also Marey, 1895, p. 228; Guidi, 1938, p. 1108). This is an ‘advantageous’ mode of flight to the insect, in that it tends to reduce the power for fixed frequency of flapping and fixed requirements of vertical force and thrust. Since the down beat *must* provide most of the force, and since the force goes up as the velocity squared of the wing while the power goes up as the velocity cubed, it requires less power for the same average force when a smaller.force is exerted over a longer time (ω_{1} < ω_{2}) than a larger force over a shorter time (ω_{1} > ω_{2}).

## GEOMETRICAL SIMILARITY AND POWER

The degree of geometrical similarity of the insects considered can be examined in Figs. 8 and 9. Here the wing length and area are plotted against the mass. The dotted line in each figure indicates the expected slope for geometrical similarity, i.e. the quantities plotted should be proportional to , respectively. Geometrical similarity is realized with considerable dispersion except possibly for the wing area, which seems to increase slightly faster than . Fig. 9 also shows that the Lepidoptera and Neuroptera (squares and circles), which tended to have smaller force coefficients, have relatively larger wing areas for their masses.

In contrast to the large individual departures from geometrical similarity for the wing area, the power as a function of the mass shows surprisingly small dispersion (Fig. 10). The power is very nearly proportional to the mass (dotted line indicates proportionality). This proportionality is reasonable from the standpoint of metabolism, and is also in keeping with the observation that the wing area increases slightly faster than . From aerodynamic considerations it can be shown that other things being equal, either the wing area must increase faster than is requires by dimensional similarity, the power increase faster than the mass, or both, as seems indicated in the present case. The uncertainty of the data renders this conclusion as yet only tentative.

Fig. 11 gives the specific power, or power per gram, as a function of the total force coefficient. They increase together, again with surprisingly small dispersion. Fig. 11 indicates about 10^{−2} horse-power per lb.—better than a man, scarcely in a class with aircraft engines !

There is a very slight negative correlation of mass with ratio of flapping velocity to velocity of flight (Fig. 12). There is also a slight negative correlation between total force coefficient and mass (Fig. 13), which does not support Rashevsky’s conclusion that *C*_{L} should increase with the mass (Rashevsky, 1944). In general, both lift and drag coefficients for insects are large, larger than those for birds, in the few cases where they have been evaluated (Demoll, 1919; Walker, 1925).

In Fig. 14 is plotted the ratio of the power expended against *F*_{p} (forces in *C*_{D}, second term of eq. (13)) to power expended against *F*_{s} (forces in *C*_{L}, first term of eq. (13)) as a function of the ratio of the root-mean-square wing-flapping velocity to the velocity of flight. Fig. 14 indicates that the fraction of the power expended against *F*_{p} increases with the importance of the inertia forces. This shows the increasing price in ‘wasted power’ paid for the use of inertia forces in flight, since *C*_{D} forces are usually in an undesirable direction, as Fig. 4 showed.

In Fig. 14 the dispersion is also surprisingly small, in view of the large individual departures from geometrical similarity. The small dispersion of quantities involving the power, as opposed to quantities involving the geometrical dimensions, indicates that as power plants they are essentially similar, despite their great diversity in form.

The range of solution for quantities involving the power (Figs. 10, 11 and 14) as the drag coefficients of the body *C*_{Db} varies between 0 and 1 is shown by the length of the vertical lines, the points being plotted for *C*_{Db}*=* 0. The dependence of the power on *C*_{Db} is considerable, a consequence of the dependence of *C*_{D} on *C*_{Db}.

There are four pairs of similar star-type points on the figures, connected by dotted lines. These are the Coleóptera. The upper point of each pair refers to calculations ignoring the elytra as wings, the lower includes them as a part of the membranous wings. This was done since the contribution of the elytra to flight was not known, but it was felt that these two cases were the limiting ones.

## REFERENCES

*Naturscissenschaften*

*J. R. Aero. Soc*

*J. R. Aero. Soc*

*Naturmssenschaften*

*J. R. Aero. Soc*

*Handbuch der Zoologie*, VII, II. Halfte. Stresemann, E., ‘Aves’

*La Locomotion chez les Animaux*. I

*Le Vol des Insectes*

*Bull. Math. Biophys*

*Nature, Lond*

*J. R. Aero. Soc*

*J. R. Aero. Soc*

By this is meant only the *mechanical* power (force × distance/time) expended by the insect. There is, of course, a much larger power loss in heating and chemical changes within the insect, as is true for any thermodynamic engine. No account is taken of this here.