## Abstract

*Physalia physalis*, commonly known as the Portuguese man-of-war (PMW), is a peculiar looking colony of specialized polyps. The most conspicuous members of this colony are the gas-filled sail-like float and the long tentacles, budding asymmetrically beneath the float. This study addresses the sailing of the PMW, and, in particular, the hydrodynamics of its trailing tentacles, the interaction between the tentacles and the float and the actual sailing performance. This paper attempts to provide answers for two of the many open questions concerning *P. physalis*: why does it need a sail? and how does it harness the sail?

## 1. Introduction

*Physalia physalis* (phylum: Cnidaria, class: Hydrozoa, order: Siphonophora, family: Physaliidae), also known as the Portuguese man-of-war (PMW for short) or the blue bottle, is a colony of numerous polyps (Totton & Mackie 1960; Bardi & Marques 2007). One of these polyps develops into a gas-filled float that looks like a sail (pneumatophore); others develop into digesting polyps (gastrozooids), reproductive polyps (gonozooids) and long hunting tentacles (dactylozooids). The float is asymmetric, with tentacles budding off-centre on approximately half of its length (figure 1). The population of the PMW is divided between those having their tentacles to the right of the sail and sailing on the starboard tack, and those having their tentacles to the left of sail and sailing on the port tack. Hydrodynamic aspects of the PMW sailing are the subject matter of this study.

To avoid ambiguity, this is perhaps the right place to define what ‘left’ and ‘right’ directions are for the PMW. To this end, consider a PMW sailing as in figure 2*a*. Were this PMW a sail boat rigged only with a mainsail, its sailing posture could have been replicated either by heading close to the wind and drifting downwind (figure 2*b*), or by sailing on a broad reach (figure 2*c*). In the first case, the sail boat is trimmed with the sail sheeted in, and a sea anchor attached to its left side, near the mast. In the second case, the sail boat is trimmed with the sail sheeted out towards the bow, and possibly no anchor at all. This sail position is rather unusual for a sail boat (it may even be inadmissible mechanically), but can be easily demonstrated on a sailboard. We believe that the first case better represents the sailing PMW, mainly because of its sail orientation relative to the body. Hence, we place the bow of the PMW at its oral end, where the tentacles are, inferring fore and aft directions accordingly. Left and right directions will be defined as if standing on the vessel and facing the bow. Thus, the PMW depicted in figure 2*a* has the sail on its right and the tentacles on its left. Totton & Mackie (1960, p. 316) defined this configuration as ‘right-handed’.

When addressing the hydrodynamics of the sailing of the PMW one cannot avoid mentioning another natural sailor, *Velella velella* (Francis 1991). The two are comparably rigged, but have very different underwater parts reflecting their feeding strategies. The PMW has a few very long tentacles that slowly drag behind the animal deep below the water surface; *V. velella* has many short tentacles that apparently work as a rake, collecting food from the water surface as the animal skims along. Although it is possible that the two have developed comparable mechanisms to harness their sails, they certainly deserve separate studies. This study is concerned with the PMW only.

## 2. A trailing tentacle

Consider a single tentacle of length *l* and uniform diameter *d* trailing through the water by being pulled horizontally with constant velocity *v* at its upper end. The forces acting on the trailing tentacle can be associated with gravity and its motion. The forces associated with gravity are given by(2.1)where *ρ*_{t}, *ρ* and *g* are the density of the tentacle, the density of water and the acceleration of gravity, respectively. The forces associated with the motion can be separated into tangential and normal components, which are assumed to be given, per unit length of a tentacle, by the constitutive relations(2.2)where *C*_{D,t} and *C*_{D,n} are the drag coefficients of the tentacle in axial flow and cross-flow, whereas *v*_{t} and *v*_{n} are the respective velocity components. *C*_{D,t} and *C*_{D,n} are estimated in appendix A as approximately 0.01 and approximately 1, respectively; *ρ*_{t}−*ρ* is estimated as a few hundredths of *ρ*.

It is shown in appendix B that under the influence of these forces the trailing tentacle remains straight. Its angle with the vertical,(2.3)is governed by a single dimensionless parameter,(2.4)representing the ratio of hydrodynamic to gravity forces (see (B 21) and (B 6)).1 The dependence of *θ* on *P* is depicted in figure 3*a*. With, say, *P*<10, minute variations in density, diameter or drag coefficient among the tentacles should cause them to spread fan-like under the float, with tentacles with smaller *P* hanging at lower angles to the vertical.

The force *T* needed to pull a tentacle is(2.5)adjusting the notation with appendix B, this equation immediately follows (B 22). *T* is depicted in figure 3 together with its horizontal,(2.6)and vertical,(2.7)components.

It is evident from figure 3*b* that the force acting on a trailing tentacle initially decreases with the trailing velocity. In fact, for small values of *P*, equations (2.3) and (2.5)–(2.7) yield(2.8a)(2.8b)(2.8c)(2.8d)An increase in trailing velocity sweeps the tentacle back (2.8*a*), increasing the vertical force component, and hence decreasing the effective ‘weight’ of the tentacle in the water (2.8*d*). Since the angle of the tentacle with the vertical is still small at this stage, the total force on the tentacle decreases as well (2.8*b*), in spite of the increase in the horizontal force component (2.8*c*).

For very large values of *P*, the tentacle becomes almost horizontal, with(2.9a)and, from that point on, an increase in trailing velocity increases the forces on the tentacle. Explicitly, for large *P*,2(2.9b)

(2.9c)

(2.9d)

In the intermediate range of *P*, there is a noticeable drop in the horizontal force *F*_{X} acting on the tentacle when *C*_{D,t}/*C*_{D,n} is smaller than, say, a few hundredths (figure 3*c*). At this range of *P*, an increase in the trailing velocity sweeps the tentacle sufficiently back to reduce the cross-flow velocity on the tentacle. Since the associated increase in the axial velocity may carry only a limited drag penalty when *C*_{D,t}/*C*_{D,n} is small, there exists a certain range of trailing velocities where the horizontal force component decreases with increasing trailing velocity. For a typical tentacle of the PMW, the ratio *C*_{D,t}/*C*_{D,n} is approximately 0.01 (appendix A), and, therefore, the decrease in the horizontal force component is expected for, say, 1<*P*<10.

Curling a tentacle increases its effective diameter and hence decreases the effective value of *P*. This obviously puts the tentacle at a smaller angle to the vertical (figure 3*a*) and reduces tension when *P* is large (figure 3*b*). Reports that the tentacles curl up when provoked (Totton & Mackie 1960, p. 376) suggest the possibility that the PMW may intentionally curl a tentacle when the tension in it crosses a certain threshold.

Using (2.3)–(2.5) and (2.1), equation (2.6) for the horizontal force component can be recast as(2.10)where(2.11)*C*_{H,t} can be interpreted as the effective drag coefficient of the tentacle; it equals *C*_{D,n} when *P* is small and the tentacle is almost vertical, and *C*_{D,t} when *P* is large and the tentacle is almost horizontal (figure 4).

## 3. Trimming the sail

The force exerted on a body moving relative to a fluid can be loosely separated into lift and drag components; the former perpendicular to the direction of the flow as seen from the body, and the latter parallel to it. In order to produce lift, the body has to be asymmetric with respect to that direction. Typically, this implies setting the body at an angle with the direction of the flow; this angle is generally referred to as the angle of attack. Indeed, in light winds, the PMW balances (trims) itself at the angle of attack of approximately 40° (Totton & Mackie 1960, p. 322). The possible ways the PMW does that are considered below.

### 3.1 Balance of forces in the horizontal plane

A schematic balance of forces in the horizontal plane is shown in figure 5. The hydrodynamic forces on the submerged part of the PMW are assumed to provide drag only and hence act in the direction opposing the direction of sailing. By definition, a point exists where these forces produce no couple. This point is called the centre of effort; it is marked ‘HCE’ in figure 5. The aerodynamic forces are generally not aligned with the wind direction, but can be associated with the respective centre of effort as well; it is marked ‘ACE’ in figure 5. In equilibrium, i.e. when the PMW is moving at a constant velocity, the aerodynamic forces are counterbalanced by the hydrodynamic forces, implying that the two are equal in magnitude and oppose each other.

The location of the centre of effort of an asymmetric lift-producing body is known to move considerably as the angle of attack changes (this result is recapitulated in appendix C—see equation (C 27) thereat). In those cases, it is more convenient to use the notion of the aerodynamic centre—a geometric point where the couple remains independent of the angle of attack (Bisplinghoff *et al*. 1996, p. 219); it is marked ‘AC’ in figure 5. The location of the aerodynamic centre is known to be practically fixed (see equation (C 28) in appendix C).

Referring to figure 5, the equilibrium conditions about the hydrodynamic centre of effort (HCE) can be written as(3.1a)(3.1b)and(3.1c)where *A*_{x} and *A*_{z} are the components of the aerodynamic force along the respective axes; *M* is the (aerodynamic) couple about the aerodynamic centre; *H* is the hydrodynamic force; *α* is the angle of attack; *β* is the angle between the direction of sailing and the downwind direction; and Δ*x* and Δ*z* are the respective distances between the aerodynamic centre of the sail and the centre of effort of the hydrodynamic forces. The associated right-handed reference frame has its *x*-axis horizontal, connecting the leading and trailing edges of the sail, and its *z*-axis horizontal and pointing to leeward.

The sailing speed *v* of the PMW (which is a few tens of centimetres per second) is small when compared with the wind speed *U* (which is a few metres per second), and hence the wind speed relative to the sail is practically *U*. Accordingly, the forces on the PMW can be conveniently represented by(3.2a)(3.2b)(3.2c)and(3.2d)where *ρ*_{A} is the density of the air; *S*_{A} and *c*_{A} are the area and the chord of the sail; *S*_{H} is an arbitrary reference area (e.g. the combined surface area of the tentacles); *C*_{x}, *C*_{z} and *C*_{M} are the respective force and moment coefficients; and *C*_{H} is the hydrodynamic drag coefficient. With these, equilibrium conditions (3.1*a*)–(3.1*c*) take on the respective forms(3.3a)(3.3b)and(3.3c)

The only parameters that may change with the angle of attack are *C*_{x} and *C*_{z}; *C*_{M} is independent of this angle by definition of the aerodynamic centre. Hence, the angle of attack at trim can be obtained as a solution of (3.3*c*). Given the angle of attack, the ratio of the first two equations,(3.4)yields the course relative to the wind; the sum of squares of these equations,(3.5)where(3.6)yields the speed.

### 3.2 Forces on the sail

The aerodynamic forces on the PMW sail can be categorized as shear forces, acting parallel to its surface, and pressure forces, acting perpendicular to it. The former contribute mainly to *C*_{x}; the latter dominate *C*_{z} and *C*_{M}, and contribute to *C*_{x}. It is shown in appendix C (see equations (C 25) and (C 29)) that for a sail with small aspect ratio,(3.7)and(3.8)where *C*_{z,α}, and are certain constants and is the ratio of the sail camber to its chord. For the course of the following discussion, the exact values of these constants are immaterial—yet it is essential that all are positive and all are of unity magnitude. Since the camber of the PMW sail is small, a few per cent chord at most, the last term in (3.7) can usually be neglected. The experimental results of Torres & Mueller (2001) suggest that (3.7) holds up reasonably well to the angle of attack of 40°. We could not find experimental results to verify (3.8).

The contribution of the pressure loads to *C*_{x} is associated with the wing surface curvature. For a conventional wing section, this contribution is loosely divided into the contribution of the leading edge region and the contribution of the rest of the wing. The former is known as the leading edge suction, and it tends to decrease *C*_{x} as the angle of attack increases. The latter is commonly accounted for as a part of the ‘parasite’ drag, which includes the shear forces as well. It changes very little with the angle of attack as long as the flow pattern about the wing remains unchanged. For small aspect ratio wings, the leading edge suction is commonly neglected, leaving *C*_{x} practically independent of the angle of attack. The upper bound of *C*_{x} is estimated in appendix A to be approximately 0.1.

### 3.3 Trim control

Since *C*_{z} changes monotonically with the angle of attack, while *C*_{x} and *C*_{M} are independent of it, equation (3.3*c*) can be rewritten as(3.9)It manifests that the side force and the angle of attack at trim are defined by the sail geometry (through *C*_{M}, which depends on the sail camber) and the relative location of the aerodynamic centre of the sail and the centre of effort of the hydrodynamic forces (through Δ*x* and Δ*z*).

On the right-hand side of (3.9), the denominator (Δ*x*) is positive by stability considerations.3 Hence, in order to obtain positive side force at trim (which is required to sail on the port tack), the numerator should be positive as well. In the numerator, *C*_{x} is positive (§3.2). Δ*z* is positive by virtue of the inherent asymmetry of the PMW's float. At the same time, *C*_{M} is a decreasing function of the sail camber (see (3.8)). Hence, there is an upper bound, , on the sail's camber.

The observations suggest that, with increasing wind speed (in particular, above 10 m s^{−1}), the angle of attack gradually vanishes (Totton & Mackie 1960, p. 322). Equation (3.9) provides three mechanisms that may explain this change in trim.

There can be a change in *C*_{M}; smaller *C*_{M} yields smaller *C*_{z}, and hence smaller angle of attack. The change in *C*_{M} is associated with the change in camber. It can come due to either (passive) aeroelastic deformation or (active) muscular contraction. There is probably no way to increase the camber passively with increasing wind speed. However, since the PMW is capable of changing its centre of buoyancy through muscular contraction (Totton & Mackie 1960, pp. 307, 374), it is plausible that the same mechanism can be used to change the camber actively.

There can be a change in *C*_{x}; smaller *C*_{x} yields smaller *C*_{z}. *C*_{x} decreases with the increasing wind speed owing to Reynolds number effects, but the associated change is estimated to be relatively small (appendix A).

There can be a change in the relative position between the aerodynamic centre of the sail and the centre of effort of the hydrodynamic forces. In particular, *C*_{z} at trim decreases as Δ*x* increases or Δ*z* decreases. The relative position of the two centres can change either due to list (roll) of the float or redistribution of the hydrodynamic drag. The former will be addressed in §4. Considering the latter, the drag is provided partly by large hunting tentacles and partly by other polyps. Drag coefficient of the large tentacles decreases with increasing sailing speed (figure 4). Drag coefficient of other polyps—which, presumably, remain as a block during sailing—should change very little as the sailing speed increases. Placing the longer tentacles aft causes the centre of effort to move forward as the sailing speed increases, increasing Δ*x*; placing them at the maximal lateral distance to the side of the sail causes the centre of effort to move towards the sail, decreasing Δ*z*.

In order to identify which of these mechanisms are relevant, consider a few representative figures. As mentioned already in §3.2, *C*_{x} is bounded from above by approximately 0.1. Judging from the available drawings (Totton & Mackie 1960, p. 310) the value of Δ*z* can probably be bounded from above by 0.3*c*_{A}. To make the numerator in (3.9) positive (see above), *C*_{M} should be greater than , approximately −0.03. At the same time, judging from plate X in Totton & Mackie (1960), the camber is non-negative, and hence *C*_{M} is non-positive. Therefore, the numerator in (3.9) is bounded between 0 and approximately 0.03*c*_{A}. But in order to obtain side-force coefficient of the order of unity, which is to be expected at the angle of attack of 40°, the denominator in (3.9) should be comparable with the numerator. Hence, Δ*x* should be a few per cent of the sail chord. These figures imply that the trim of the PMW is a delicate one, and a minute shift of the centre of effort of the hydrodynamic forces or a minute change in the sail camber may change it completely.

Detailed description of the cormidia found in Totton & Mackie (1960, pp. 340–346) places the long tentacles aft and away from the sail. It implies that the passive trim control based on the tentacles' drag may be in use by the PMW. An indirect support to this conclusion is provided by the sailing speed of the PMW. In fact, the major part of the transition between the high and low angles of attack at trim should be completed when *P* of its hunting tentacles becomes of the order of 100 (figure 4). With *d*=2 mm, *C*_{D,n}=1 and *C*_{D,t}=0.01, *P* equals 100 at approximately 0.3 m s^{−1} when the tentacles have a negative buoyancy of 3 per cent, and approximately 0.4 m s^{−1} when they have a negative buoyancy of 5 per cent. These figures are comparable with the maximal sailing speed of approximately 0.4 m s^{−1} reported by Totton & Mackie (1960, p. 318).

Having the long tentacles positioned away from the sail and relatively aft infers that, when they capture prey, their effect on the trim is just the opposite of what has been previously discussed—an increase in their drag changes the trim towards higher angles of attack and hence amplifies the force on the tentacles. In order to prevent tearing the tentacles off and to allow pulling the prey towards the digesting polyps, the sail has to be ‘sheeted out’ (luffed). We believe that the PMW has the option to luff its sail by increasing the sail's camber through muscular contraction of the float. A change of the order of 1 per cent chord—a couple of millimetres—should suffice to this end (see equation (3.8) and the paragraph immediately following it).

Passive speed control has no counterpart in the sailing world, where most autopilots are designed to keep the course, either relative to the wind or relative to the Earth, rather than to keep the speed relative to the water. Yet, presented with the requirement to keep the speed when drifting on a sea anchor, as in figure 2*b*, we can hardly think of a simpler design than that mimicking the PMW. Tie the sea anchor (representing here all the polyps other than the hunting tentacles) relatively forward on the port side, and hang a suitable non-buoyant long cable on the same side but astern. As the drift velocity increases, the cable will lift up, heading the boat into the wind and easing the sail.

## 4. List

As the angle of attack is a consequence of the balance of forces in the horizontal (*x*–*z*) plane, the roll angle (list) is a consequence of the balance of forces in the vertical (*y*–*z*) plane. A schematic balance of forces in the vertical plane is shown in figure 6. The horizontal side force on the sail, *A*_{z}, is counterbalanced by the respective component, *H*_{z}, of the hydrodynamic drag. The weight of the float, *W*, and the effective weight (weight minus lift) of all other polyps, *H*_{y}, are counterbalanced by the (mainly hydrostatic) lift of the float, *B*.

The point where the lift acts is marked ‘CB’ in figure 6. Listing to leeward moves this point to leeward as well. The list stabilizes when the couple produced by the lift counterbalances the couple produced by the aerodynamic forces on the sail–float and the hydrodynamic forces on all other polyps. Obviously, increasing *A*_{z} will increase the list to leeward; conversely, decreasing *A*_{z} will change the list to windward.

List to leeward increases Δ*z*, and hence increases *A*_{z} by (3.9); in turn, increasing *A*_{z} increases the list. This is exactly what happens when a sudden gust hits. Until the tentacles lift up and readjust the trim (it should take about the same time it takes the PMW to sail the length of its tentacles, approx. 30 s), the increasing force on the sail increases the list, which increases the angle of attack, which increases the list further (Totton & Mackie 1960, p. 322). This scenario reverses when the wind drops. A sudden lull decreases *A*_{z}, which changes the list to windward, which decreases *A*_{z} even further, possibly leading to a capsize (Totton & Mackie 1960, pp. 309, 374). A consequence of this amplification of the list is the existence of minimal wind strength allowing the PMW to use its sail.

Capsizing appears as a single obstacle that prevents the PMW from changing tacks (Totton & Mackie 1960, p. 321). In fact, by increasing the camber of the sail, the numerator in (3.9) can be made negative, resulting in negative *A*_{z}. Negative *A*_{z} implies sailing on a starboard tack.

## 5. Course and speed

The course of the PMW relative to the wind is determined solely by the angle of attack at trim. From (3.4)(5.1)The largest angle from downwind, *β*^{*}, is obtained at the angle of attack *α*^{*}, which is the solution of d*β/*d*α*=0. Since *C*_{x} was assumed already to be independent of *α*, differentiating the right-hand side of (5.1) with respect to *α* yields(5.2)

With *C*_{z} given by (3.7), the solution of (5.2) is(5.3)at which(5.4)

Consider typical figures. As already mentioned above, *C*_{x} is of the order of 0.1; and are about unity; is a few hundredths, and hence insignificant. Thus, is approximately 0.3, corresponding to *α*^{*} of 15–20° and *β*^{*} of 50–55°. In comparison, at the angle of attack of 40°, equation (5.1) yields *β* of 40–45°, about the value reported by Totton & Mackie (1960, p. 321). This means that, by reducing the angle of attack, the PMW could have achieved a better course relative to the wind. It is therefore plausible that the course relative to the wind was not the aim of the PMW's design. Rather, its design was probably aimed at keeping the pull of the sail as constant as possible (independent of the wind speed) so as to allow the best possible conditions for the tentacles to spread out.

In order to estimate the sailing speed, data are required concerning the hydrodynamic drag *H*. It includes the drag of long (hunting) tentacles, *H*_{t}, and the combined drag of all other polyps, *H*_{p}. For simplicity, the drag of the last group will be approximated with a simple constitutive relation(5.5)where *S*_{p} is a suitable reference area; *C*_{H,p} is the associated drag coefficient; and all remaining quantities pertain to the average tentacle. The drag of *N* tentacles will be approximated as *N* times the drag of the average tentacle, i.e.(5.6)see (2.10), (2.4) and (2.11). With (5.5) and (5.6), equation (3.5) for the sailing speed can be recast as an algebraic equation for *P*,(5.7)

We assume that a representative (nominal) PMW (appendix A) has sail area *S*_{A} of 0.01 m^{2} and seven tentacles; the average tentacle is 10 m long, 2 mm in diameter, has a negative buoyancy of 3 per cent and is characterized by *C*_{D,n}=1 and *C*_{D,t}=0.01. These set its weight in the water, *F*_{g}, at approximately 1 g. The drag area of its digesting and reproductive polyps, *S*_{p}*C*_{H,p}, is guessed as 0.001 m^{2}. With these, the solution of (5.7) is shown in figure 7. At each value of *C*_{z}, the sharp increase in the sailing speed corresponds to the lifting of the tentacles towards the horizontal—these are exactly the conditions where the spreading of the tentacles is the most effective. The observations of the sailing speed reported by Totton & Mackie (1960) are marked by the circles.4 They inferred that *C*_{z} decreases as the wind speed increases (i.e. the PMW ‘sheets out’ its sail), and that the stronger winds are hardly optimal for spreading the tentacles.

## 6. An example

To elucidate the trim of the sail, consider now a more elaborate model of the PMW. This artificial model is entirely made up to serve as an illustrating example. In this model, we guess the centre of effort of the hydrodynamic forces acting on the block of digesting and reproductive polyps, differentiate the seven long tentacles by length and diameter, and specify their connection points with the float. The geometry (mimicking the PMW depicted in figure 8) is shown in figure 9; the associated numbers can be found in appendix A.

To simplify the solution of (3.5) we invert the approach of §5 and assume the sailing speed, *v*, rather than the wind speed, *U*, to be the independent variable. Given *v*, *P* for each tentacle immediately follows from (2.4). The tilt angles of the tentacles follow from (2.3); the associated drag forces follow from (2.10); the drag of all other polyps follows from (5.5). The knowledge of the drag forces allows the HCE to be found; the trim follows from (3.3*c*). Once the trim is known, the wind speed and the course relative to the wind follow from (3.5) and (3.4). The results of this procedure are shown in figure 10, once for *C*_{M}=−0.01 (solid curves) and once for *C*_{M}=−0.023 (dashed curves).

The ‘L’-shaped track of the HCE in figure 9 is an immediate consequence of our placing the thin (and hence the fastest to lift up) tentacles forward and close to the sail. As the speed increases and the thin tentacles begin lifting, the centre of effort moves aft; once the bigger tentacles (located to the side of the sail) begin lifting, it moves inward. We remind readers that this is just an example and this track could have been shaped at will by altering the diameters and the lengths of the tentacles, their connection points, etc.

The angular spread of the tentacles is biggest in light winds (figure 10*c*). The wind speed at which this best spreading occurs is influenced by many parameters of the model, conspicuously by *C*_{M}, but also by the negative buoyancy of the tentacles and their diameters. *C*_{M} controls the angle of attack; the buoyancy and the diameters control the sailing speed at which the tentacles lift up (through *P*).

The sailing speed rapidly increases in light winds, from practically 0 at 2 m s^{−1} to more than 0.1 m s^{−1} at 2.5 m s^{−1} (with *C*_{M}=−0.01). This behaviour was reported by Totton & Mackie (1960, p. 322); it is an immediate consequence of the tentacles losing drag as they lift towards horizontal (§2). A minute decrease in *C*_{M}, from −0.01 to −0.023, corresponding to an increase of less than 1 per cent chord in camber, is sufficient to luff the sail almost completely—compare solid and dashed curves in figure 10*d*.

## 7. Concluding remarks

The most conspicuous results of this study can be summarized in three points as follows.

The tilt angle (from vertical) of a trailing tentacle increases with the sailing velocity and decreases with the tentacle diameter. Since the tentacles are of different diameters, trailing causes them to spread fan-like in the vertical plane. The spread angle vanishes both at small and at large trailing speeds. Using plausible figures for the diameters, densities and drag coefficients of the tentacles, the spread angle is estimated as 10–15° at the same speeds the PMW were observed sailing.

The force needed to trail a tentacle at a given speed decreases as the tilt angle of the tentacle increases. The tentacles bud from the float in such a way that this change in drag sheets out the sail as the tentacles lift up, which serves to keep the sailing speed in that particular range where the tentacles spread best. The PMW were observed sailing with their sails aligned with the wind in strong winds.

The trim of the sail is very delicate and a minute increase in camber may luff the sail.

We believe that these results can be rationalized by adopting a hypothesis that the PMW has evolved for efficient sail-driven trawling: the tentacles' spreading obviously increases the likelihood of catching the prey (Madin 1988; Purcell 1997), whereas the PMW ‘does’ all it possibly can to keep them spread.

We hope that this study will encourage new observations and measurements that will allow for a better understanding of this singular colony. A partial list of missing data includes tentacles' dimensions, density and attachment points to the float, float geometry and drag of all polyps which are not the hunting tentacles.

## Footnotes

↵Sin

*θ*can be identified with used in appendix B;*P*can be identified with the product*pC*_{D,n}.↵This is not a formal expansion—it includes the leading order term (with respect to

*P*) containing the ratio*C*_{D,t}/*C*_{D,n}, and the leading order term which is independent of it.↵Stability of a trimmed sail is manifested in maintaining its angle relative to the wind—a sudden increase in the angle of attack should produce a restoring couple. In other words, the derivative of the aerodynamic couple about HCE,

*C*_{M,H}=*C*_{M}+*C*_{x}Δ*z*/*c*_{A}−*C*_{z}Δ*x*/*c*_{A}, with respect to the angle of attack should be negative.*C*_{x},*C*_{M}, Δ*x*and*c*_{A}are independent of the angle of attack, the first by assumption (§3.2), the others by definition; Δ*z*slightly increases with the angle of attack through increasing roll (§4), whereas*C*_{z}increases monotonically with the angle of attack. Hence, for the derivative of*C*_{M,H}to be negative, Δ*x*should be positive.↵The two higher speed observations are reported in full on their p. 319. The lower speed observation, appearing on their p. 322, lacks the actual sailing speed. We estimated it from the course direction (40° from downwind) and from the normal to the wind velocity component of 0.13 m s

^{−1}(1/4 knot).↵Just as any winged slender body configuration (Ashley & Landahl 1985, p. 122).

↵Pertinent equations in Raymer (1992) are (12.24), (12.25), (12.27) and (12.31).

↵Pertinent equations in Raymer (1992) are (12.27) as (12.28).

↵In notation of appendix C, .

↵For a sail, protruding perpendicular to an impermeable water surface, aspect ratio is defined as twice the height of the sail squared divided by its area (appendix C).

↵In notation of appendix C, .

- Received October 25, 2008.
- Accepted November 17, 2008.

- © 2008 The Royal Society