360 
NATURE 
[May 27, 1915 

which it is seen that for a naked boiler, 7.e. one with- 
out a glass cover, the maximum oyerall efficiency 
obtains when T=692° F. corresponding with a steam 
pressure of 21 lb. sq. in. abs. Or, of course, if equa- 
tion (3) be differentiated with regard to T, and the 
result equated to 0, we also obtain T=692° F., from 
which is obtained the maximum value of 7). 
THE RIGID DYNAMICS OF CIRCLING 
PLIGHT 
NE of the main objects of the investigation 
described in this paper is to ascertain the condi- 
tions which render it easiest to steer an aeroplane in 
a horizontal circle of any radius that is not too small, 
and the idea of “inherent controllability’? has been 
introduced to denote the property which a system may 
possess of freely describing a circular path without 
any pressure on the controlling rudders. In such 
cases the rudders will act as guides by preventing 
the aeroplane from leaving the chosen path, and as 
the system without them must, from the nature of 
the case, be wanting in directional (lateral) stability 
it is necessary for the working of such a system that 
the addition of the rudders should render it stable. 
The conditions for this must be worked out by the 
methods described in the lecturer’s book, ** Stability in 
Aviation.” 
It will be found that there are several different 
ways of obtaining inherent controllability and that 
in circling flight the system turns about a point which 
in some cases is in front and in some cases behind the 
centre of gravity. The axis of x or horizontal line 
through the centre of gravity in the direction of 
forward motion thus envelopes a circle of radius, a, 
the ‘‘turning point” beiag the point of contact of 
the axis tangent with its envelope, and the’ lateral or 
sideways velocity of the aeroplane being proportional 
to the distance of the centre of gravity from the 
turning point. This length and the inclination of the 
aeroplane to the vertical constitute two independent 
variables which can be so chosen as to satisfy two 
conditions of lateral equilibrium, but as there are 
three, a third variable is in general required, and if 
a rudder plane is used, this latter variable may be 
taken to be the pressure on that plane. The condition 
for inherent controllability is that the three equations 
of lateral equilibrium should satisfy some further 
identical relation by which the number of variables 
is reduced to two, and there are several ways in 
which this may be done. 
The results here summarised lead to some interest- 
ing conclusions which were quite unexpected when 
the paper was commenced. In particular, they show 
the differences in behaviour between wings that are 
bent up and down respectively, the advantages, in 
certain circumstances, of curved wings as contrasted 
with plane wings bent into a simple dihedral angle, 
and generally that the form and curvature of the 
wing areas may play a much more important part in 
circling flight than had been anticipated. 
The applications to the flight of birds are obvious, 
and suggest much interesting material for discussion. 
At any rate, a good many peculiarities in the wing 
structure of the circling birds appear to admit of 
interpretation on dynamical principles. 
With regard to the possible application: of these 
results to actual areoplanes, it remains to be seen 
how. far it is desirable or practicable to realise the 
conditions of inherent controllability in a real flying 
machine. But the lecturer suggested that a study of 
1 Abstract of the third Wilhur Wright Memorial Lecture delivered before 
the Aeronautical Society on May zo, by Prof. G. H. Bryan, F.R.S. 
NO. 2378, VOL. 95| 


the present work, followed by a few experiments, will 
either lead to improvements in the steering of aero- 
planes, or if the present arrangements are the best, 
it will now be easier to understand the reason why. 
Summary and Conclusions. 
(1) In steady motion in a horizontal circle, both 
the longitudinal and the lateral equations of equili- 
brium are affected. 
(2) The turning point may be in front of or behind 
the centre of gravity, its distance when in front being 
denoted here by b. 
The axis of the aeroplane then envelopes a circle 
of a certain radius a, the real radius of the circle 
described being /(a?+)’*). 
The system usually cants over sideways through a 
certain agle 9. 
(3) Given the velocity and radius of the circle it is 
not usually possible to satisfy the three equations of 
lateral equilibrium by assigning suitable values to 
b and g, but when this is possible the system is said 
to be inherently controllable. 
In an inherently controllable system the rudder 
planes merely act as guides, and it is necessary ‘that 
they should be so placed as to render the motion 
laterally stable. 
In other cases steady motion can ‘only be main- 
tained by pressure exerted by the rudders or a couple 
applied by means of ailerons or some such action 
representing the third unknown variable required for 
the solution of the three simultaneous equations of 
lateral equilibrium. 
(4) In a system of straight planes sing is propor- 
tional to the radius a of the envelope, but it also 
appears that the other conditions of lateral equilibrium 
are only possible when pressure is applied.by means 
of a rudder, and when a and @ have certain definite 
values. -The only way of varying the radius of the 
circle actually described is by varying the position 
of the turning point, which may be in front of or 
behind the centre of gravity. 
The addition of boxed-in ends or vertical partitions 
improves the steering, but it still leaves sing pro- 
portional to a. The inference one would naturally 
derive from the formula is that all such systems 
would be liable to sway from side to side of the 
straight path in curved arcs of finite radius. In no 
case can the radius of the circular envelope exceed 
the limit corresponding to »=g0°. 
(5) With bent-up wings, as in the ‘Antoinette 
type,’’ it is possible to satisfy the conditions of equili- 
brium so that a is no longer limited and @ no longer 
large. Such a system can be steered in a circle of 
large radius without being inclined at a large angle. 
In general, circular motion can only be maintained 
when pressure is applied by means of a rudder or a 
couple applied by means of ailerons, but if the two 
principal moments of inertia about axes perpendicular 
to the line of flight are equal, the rudder exerts no 
pressure, and the system is inherently controllable, 
the inclination satisfying the relation U*=ga tan 9. 
(6) Another kind of ‘‘inherent controllability’? in 
which the system always remains level, the inclination 
being zero, is possible in certain systems. A neces- 
sary condition is that the wings should be bent down- 
wards and not upwards at the tips, and it will be 
usually advantageous that they should be most bent 
down at their extremities. The condition represent- 
ing this fact is that the space between the wings 
and a chord joining their tips should be as large as 
possible. 
This arrangement of the wings somewhat repro- 
duces the action of gulls’ wings in circling flight, 
and it will be found that differences in the form and 


