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BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY::
SURFACES OF REVOLUTION OF MINIMUM
BY DR. E. J. MILES.
(Read before the Chicago Section of the American Mathematical Society,
January 1, 1909.) The problem of finding the surface of revolution of minimum resistance may be thought of as the oldest problem of the calculus of variations. A first solution was given by Newton* in 1686. It has since been considered by L'Hospital, August, Silvabelle, Kneser, and others. The results obtained by these writers are based on the Newtonian law of resistance, which states that the resistance R is given by the formula
R = f. sina, where f is the force and a the angle which the line of force makes with the tangent to the surface at the point of application. However, physical experiment does not always verify this law. Especially does it failf when the angle a is small.
As a result of this, several different laws of resistance have been given; some being derived mathematically, others being stated as verifying experiment. Among these the laws of von Lössl, I Duchemin, $ and Kirchhoff have received the greater notice. They are given by the following formulas:
(4 + ) sin a
4+ a sin a * See Principia Philosophiæ Naturalis, II; Sect. VII, Prop. XXXIV, Scholium.
† For an account of the various physical causes underlying this, see Encyklopädie der mathematischen Wissenschaften, IV, 17, $$ 4, 5, 6.
IF. v. Lössl, Die Luftwiderstandsgesetze, Wien, 1896, p. 96.
§ Duchemin, Experimentaluntersuchungen über den Widerstand der Flüssigkeiten, Braunschweig, 1844, p. 101. This law has been verified by Langley, Experiments in Aërodynamics, Washington, 1891, p. 101.
|| G. Kirchhoff, Journal für Mathematik, vol. 70 (1869).
2 sin a
The presont paper had its origin in a study of the surfaces of revolution resulting from these separate laws. In each case it was found that the curve determining the surface had properties quite analogous to those of the newtonian curve. This nåturally suggested the determination of a general law of resistance having properties which would include the preceding ds special cases. Such a law is stated in $1. In this section the discussion of the properties of the curves resulting from this law is given. In the following three sections the theory of the special cases will be taken up in the order mentioned as applications of the general. This general case also includes the newtonian law of resistance and from it the well known results* for this law can be obtained.
§ 1. Surface of Revolution of Minimum Resistance for a
General Law of Resistance. In formulating the problem it will be supposed that the surface is formed by the revolution of an arc AB x = x(t), y = y(t)
(t1 St=ta), about the X-axis, where the point A is taken at the origin and B in the interior of the first quadrant. The direction cosines of the tangent are then given by the expressions x'(t) and y'(t). Let this arc be such that y> 0 except for t = tı. Further it will be supposed that the body moves in the direction of the negative X-axis with constant velocity.
Consider then the surface resulting from the law of resistance expressed by the formula (1)
R= f.(x', y'), where the function y(x", y') is homogeneous and of dimension zero in x' and y', and f again represents the force. It is readily shown that, aside from a numerical factor, which is independent of the form of the curve, the resistance is given by the definite integral (2)
yy' (x', y')dt.
However from physical considerationst it is necessary to limit
* For a very complete summary of the work done on this classical problem see Bolza, Vorlesungen über Variationsrechung, pp. 407-418.
+ See Áugust, Journal für Mathematik, vol. 103 (1888), p. 1.
the discussion to curves such that along the arcs AB one has
x = 0, y'20. Consequently the problem under consideration may be stated thus:
Among all ordinary curves which go from the origin A to à point B given in the interior of the first quadrant and which besides satisfying the regional restriction (3)
y> 0 for ti < t = tz also satisfy the slope condition (4)
2' 20, y' 20 forti et sta, it is required to find that one which minimizes the integral (2).
Since the restriction has been made that the expression (z', y') is homogeneous and of dimension zero it follows that it can be written as a function of q, say 4(q) for simplicity, where (5)
q= /y' = cot 0,
e being the angle which the tangent makes with the positive X-axis.
The following restriction will now be imposed on the function 49):
Its derivative o' (9) first decreases continuously from zero to a finite minimum value and then constantly increases to the value zero as q increases from 0 to + ..
Graphically the function (9) is as shown in Fig. 1. If c denotes the value of q, when (q) has its minimum, then ''(g) assumes all values between 0 and 6 (c) twice.
These conditions are fulfilled in all the special cases to be considered.
Suppose now that an arc of the minimizing curve is considered which is of class C' and such that
This arc must then satisfy the Euler differential equation, from which it follows that a first integral is given by the equation (7)
yy'd y(x", y')/ax' = yo'q) a,
where a is a constant of integration which must be positive on account of the above conditions. Hence it follows that
But from the relation (5) x' is seen to have the value
x' = a
aX(q) + b, y = aY(q), furnish the most general solution of the Euler differential equations in terms of the parameter q, when the inequality (6) is satisfied.
But in as much as the general extremal can be obtained from the special curve
1 O'(9) (9) X = X(Q) q) = qdq, Y = Y(Q)
by means of the similarity transformation