Art. LVIII.—On the Mechanical Principles involved in the Flight of the Albatros. * This and the following paper on “Sinking Funds” (Art. LX.), had to be reserved last year, for want of the necessary algebraic type: they are now printed, together with a reply by Mr. J. S. Webb, to Captain Hutton's paper on the “Flight of the Albatros.” As it was still found impossible to procure all the mathematical signs, the following substitutions have been made throughout:— For Greek Beta the letter F has been inserted, "Theta " I " "Phi " Q " —ED.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 2, 1869, Page 227

Art. LVIII.—On the Mechanical Principles involved in the Flight of the Albatros. * This and the following paper on “Sinking Funds” (Art. LX.), had to be reserved last year, for want of the necessary algebraic type: they are now printed, together with a reply by Mr. J. S. Webb, to Captain Hutton's paper on the “Flight of the Albatros.” As it was still found impossible to procure all the mathematical signs, the following substitutions have been made throughout:— For Greek Beta the letter F has been inserted, "Theta " I " "Phi " Q " —ED. By Captain F. W. Hutton, F. G. S. [Read before the Auckland Institute, June 1, 1868.] Perhaps no subject in ornithology has been less satisfactorily treated than that of flight, although it possesses very great interest, both for the naturalist and the mathematician. It is, however, one of considerable difficulty, as it has to deal with the complicated question of the resistance of the air to bodies moving with variable velocities; and the following remarks do not pretend to do more than indicate the principles involved in the flight of the albatros when sailing along without moving its wings. I must premise, at starting, that I take it for granted that no movement of the wings, body, or feathers of the bird takes place other than those necessary for seeking its food, or altering its direction of flight, as all observers are agreed on this point. It may also be necessary to remark that the velocities spoken of are velocities of the bird through the air, and not over the water, which might be very different. For example, suppose an albatros to be flying with a velocity of 40 feet a second, against a wind having also a

velocity of 40 feet a second; it is clear that the bird would remain stationary with regard to the sea over which he was flying, nevertheless he would have a velocity of 40 feet a second through the air, just as much as if the day was quite calm, and he was flying both through the air and over the water equally at the rate of 40 feet a second. This being understood, I will first suppose an albatros, on a perfectly calm day, to be placed in the air at some distance above the sea-level, with its wings and neck stretched out in the attitude of flight, but without any forward movement. It is clear that the moment the support was withdrawn, it would commence falling in a nearly vertical direction, unless, indeed, it had power to buoy itself up by inflating its air-cells with hot air. That it has not this power I have elsewhere shown (See “Ibis,” July, 1865), but for the sake of completeness I may perhaps be allowed to repeat it here. “The temperature of the albatros, as taken by Sir G. Grey, by placing a thermometer under the tongue, is 98° F., and if we add 10° F. to this, in order to allow for the difference between the head and the body, we shall have the temperature of the air-cells at 108° F. The temperature of the surrounding air cannot be taken lower than 48° F.; the bird, therefore, could not raise the temperature of the air taken into these cells more than 60° F. This would increase its volume not quite one eighth; and taking 100 cubic inches of air to weigh 31 grains, and the average weight of an albatros to be 17 lbs., it would be necessary, in order that the specific gravity of the bird might be brought to that of the atmosphere, that these cells should contain 1820 cubic feet of air, or in other words, they must be more than 1000 times the size of the body of the bird. In fact it would require a sphere of more than 15 feet in diameter to contain the necessary quantity.” This objection being disposed of, it follows that the bird must fall. If now we take the area of the under surface of the body, neck, and expanded wings and tail of the albatros to be 8 square feet, and its weight 17 lbs., we see that it would take an upward pressure of 2–12 lbs. per square foot, to support it. This pressure would be given by a current of air moving upwards with a velocity of 20 feet a second, so that on a perfectly calm day the bird would fall downwards at a constantly increasing rate, until it had attained a velocity of 20 feet a second, which velocity it would keep until it fell into the sea. This is called its “terminal velocity.” It is necessary to notice that as the position of the body, wings, and wing-feathers of the bird would be inclined to the horizon, the direction of descent would not be quite vertical, but would be inclined in the same direction as the body and feathers of the bird, namely, backwards. I will next suppose that instead of being calm, a breeze is blowing with a velocity of C feet a second. The bird would now, of course, be forced back in the direction of the wind at the same time that it was falling. But it is well-known that when a body at rest is set in motion, by a force acting upon it, the body commences to move gradually, and acquires a certain velocity in a certain time which is represented by the formula v. = P. g. t./W    (1) where v. is the velocity acquired in the time t., when a force P. acts upon a body weighing W. lbs., g being the force of gravity. This is called the “inertia” of the body. If then we suppose the bird to be facing the wind, the backward velocity, communicated by the wind would increase, while the force of the wind upon it would decrease, but of course P+v would always be equal to C. ∴ v= C-P   (2)

combining these two equations we get t= W (C-P)/g P now when the velocity of the bird equalled that of the wind, P. would be o, in this case ∴ t= C W/o =∞ so that the bird could never actually acquire a velocity equal to that of the wind, and there would always be a force of C—v. acting on it, and as the bird, its wings and its feathers, would be inclined at an angle, which I will call Q, to the horizon, and therefore to the direction of the wind, this force would be resolved into two—one, equal to (C—v) sin2 Q, tending to drive it backwards —and the other, equal to (C—v) sin Q cos Q, tending to delay its fall or even to raise it, supposing it to be sufficiently great to overcome the force of gravity. But even in this case C—v. is constantly decreasing as it approaches its limit v=C., so that there must always come a time when (C—v) sin Q cos Q is not sufficient to support the bird, and it must commence to fall; so that in all cases it would reach the water in a curved line at a certain distance behind the first position of the bird, the form of the curve depending on C., W., and Q. I have dwelt thus minutely on these simple facts, because it has been supposed that in a gale of wind, a certain position merely of its wings or feathers, might enable an albatros to sail against the wind, without any momentum of its own, which is quite impossible. Another explanation that has been given is that the albatros can fly almost against the wind, in the same way that a ship beats to windward. This, however, is manifestly incorrect. A ship is placed in two different media, one of which—the water—is practically stationary, and it is enabled to sail at an acute angle with the wind, because the pressure of the wind, being met by the resistance of the water, is resolved into forces having other directions, and advantage being taken of this by trimming the sails, it ultimately results that the ship is moved in the direction of least resistance, viz., forwards. But the case of the albatros presents no analogy to this; it is placed in one medium only—the air, the whole of which is moving in the same direction, and with the same velocity, and it has no means, unless by using its wings, of offering any resistance, except its inertia which we have already seen is not sufficient, to the wind and so resolving its direction into others more advantageous to itself, in fact it is analagous to a balloon, which, except by the aid of machinery, can only drift with the wind. Having, therefore, seen that while the wings are stationary, no forward movement can be commenced by the bird, we are forced to the conclusion that the albatros sails along by means of the momentum that he had previously acquired by strokes of his wings on the air, or of his feet on the water when rising from it, or from both combined, and that so soon as the resistance of the air has reduced his velocity so much that it no longer prevents him falling, fresh impulses of the wing have to be given. It will now be observed that the difficulty has been shifted from the means of obtaining motion through the air, to that of keeping up a velocity but slightly diminished, for so long a time as the albatros is known to sail without using his wings, or in other words, to the very small resistance that the air must offer to his progress; and if it could be shown that this resistance is not too great to allow for the longest observed time of sailing, all difficulties with respect to this part of the flight of the albatros would disappear. I do not profess to have done this, but I think that I can show that there is no insuperable difficulty in the way.

Suppose now the body of the bird to be inclined at an angle Q to the horizon, and moving through the air with a velocity of v. feet a second, it would rise (omitting the force of gravity for the present) by the angle at which it was flying V. tan Q feet a second, and the resistance of the air to its wings would give it a further upward movement of v sin Q cos Q feet a second, so that the total rise of the bird would be v (tan Q+sin Q cos Q) feet per second; but we have already seen that the terminal velocity of the bird is 20 feet a second, so that in order to find the velocity at which the albatros must fly at an angle of Q to the horizon in order to make the upward movement just sufficient to counteract the force of gravity, or in other words to maintain a horizontal line of flight, we have v (tan Q + sin Q cos Q) = 20 ∴v = 20/tan Q + sin Q cos Q = 20 cos Q/sin Q (1 + cos2 Q) If we take Q to be 5°, we shall find that v=116; and if we take it to be 10°, we get v=58. It appears, therefore, that if an albatros starts with a velocity of 116 feet a second, while sailing at an angle of 5° to the horizon, he could maintain a constant height above the sea level until his velocity was reduced to 58 feet a second, by gradually increasing the angle at which he was flying to 10°. I will now compare the actual known resistance of the air to a round shot, to what ought to be the resistance to an albatros to allow it to sail for half an hour without using its wings, and only reducing its velocity from 116 feet to 58 feet per second. The formula for calculating the resistance to round shot as given by Parcelet, is R=0.0006 A v2 where A is the resisting area in square feet. If now we take the area offered to the air by the front surface of the bird to be 0–66 square feet, and its mean velocity at 87 feet per second, we have by the round shot formula R = 0.0006 × 0.66 × 872 =3 lbs. nearly, which is evidently too great. I will now estimate roughly the real resistance the albatros ought to have met with in order to enable it to sail for half an hour. Taking the average velocity at 87, it is evident that in half an hour it would traverse 156,600 feet. Now at starting it would have accumulated W/2g v2 = 17 × 1162/2 × 32 = 3599 units of work, at finishing it would still have unexpended 17 × 582/2 × 32 = 900/ units of work. So that substracting one from the other, 2699 units of work have been consumed in going 156, 600 feet, and the resistance overcome would be 0–017 lb. per foot, or only 1/176 of that calculated by the round shot formula. This, how-

ever, gives rather too small a result, as the average velocity must be under 87 feet a second, and I will try a more correct way of arriving at the result. Let w be the weight of the bird in lbs., V its velocity at starting, and v its velocity after having sailed over s feet in t seconds; and let v′ be its velocity after having sailed over s′ feet in t′ seconds. If now we suppose the time between t′ and t to be very short we may assume the resistance of the air to be constant throughout the small space s′—s, and to be equal to × Av2. Therefore W/2g v2 = W/2g v′2 + × Av2 (s′—s). W/2g (v′2—v2) = -x Av2 (s′′s) Or, W/2g. dv2/d s = -x Av2 ∴ dv2/ds = -2 Fv2 where F represents g Ax/W. ∴ v2= Ce−2Fs Now when s=o, v=V ∴ C=V2 And we obtain v2 =V2 e−2Fs v=Ve−Fs eFs=V/v   (1) But when the time is very short we may suppose that the velocity of the bird would remain the same throughout it, and therefore s′—s=v (t′—t) Or ds/dt =v= Ve−Fs ∴FVt=eFs + c Now when t=o, s=o ∴ c= -1 And eFs FVt + 1   (2) Equating this result with that obtained in equation (1) we get, V/v =FVt + 1 F=(V/v-1) 1/Vt =V-v/Vvt Substituting gAx/W for F we have gAx/W= V−v/Vvt ∴ x=W(V–v)/VvtgA.

Therefore in the case that we are supposing with the albatros x=17 × 58/116 × 58 × 1800 × 32 × 0.66 ∴ x=0.000004 nearly. And the resistance to the bird would be R=0.000004 Av2 Which is 1/100 of the resistance as calculated for round shot. This difference seems very great, but several things have to be taken into consideration. In the first place the resistance obtained for the albatros is calculated on the supposition that both its under and front surfaces were planes, which is far from being the case. The under surface of the wings is concave, and perhaps offers three times the resistance of a plane surface, which would greatly reduce its terminal velocity, and therefore both the velocity at which the bird was compelled to fly in order to maintain its height above the sea, and the resistance offered to its forward movement. On the other hand the front surface is very well adapted for piercing the air, and as the resistance to a round surface is only about one-third of that to a plane, and to an elongated shot only one-sixth of that to a round shot, we might fairly presume that these two together might reduce the resistance to one-fiftieth part of that calculated for round shot. Again we must remember that this result is obtained by supposing that the law, as determined for the velocities of round shot, holds good for lesser velocities, or that the resistance always decreases as the square of the velocity; but it is well known that this is not strictly the case even with high velocities, and it is probable that the law is very incorrect when the velocities, and shapes of the bodies, differ very considerably. For example, the range of an ordinary round shot starting with an initial velocity of 1200 or 1600 feet a second, can be pretty accurately calculated by the formula here used, but in the case of a mortar-shell, starting with an initial velocity of only 300 or 400 feet a second, the range is much better obtained by the parabolic theory, which omits the resistance of the air altogether, than by Parcelet's formula; and the velocity of the albatros is small, even when compared with that of a mortar-shell. The actual resistance of the air to the bird can only be determined by accurate experiments, and it is important that they should be taken, as until they are completed no satisfactory conclusion can be arrived at with respect to flight. From the foregoing observations we can easily understand how it is that the albatros never sails for long in calm weather, for when no wind is blowing, its velocity over the water would be as great as that through the air, and it would have to rush along so fast that it could not search the sea properly for food, nor stop itself quick enough when it saw anything. I have thus endeavoured to point out what appears to me to be the only possible way of accounting rationally for the wonderful flight of the albatros, but once more I wish it to be understood that I by no means pretend to have solved the problem, but only to have cleared the way for solving it. Experiments are required for determining accurately the resistance offered both by the front and under surfaces of the albatros to different velocities of wind, and if I should ever be in a position to undertake these, I shall not fail to lay the results before the members of the Institute.