Owing to certain theorems that have been discovered by Lindelöf and other writers, some of the very characteristics of a minimal surface of rotation, which are sought in the Calculus of Variations, may be obtained for the case of the revolution of the catenary without the use of that theory. We shall give these results here, as they offer a handy method of comparison when we come to the results that have been derived through the methods of the Calculus of Variations.
In presenting the subject-matter of this Chapter, the lectures given by Prof. Schwarz at Berlin are followed rather closely. The results are derived by Todhunter in a somewhat different form in his Researches in the Calculus of Variations, p. 54; see also the prize essay of Goldschmidt, Monthly Notices of the Royal Astronomical Society, Vol. 12, p. 84; Jellett, Calculus of Variations, 1850, p. 145; Moigno et Lindelöf, Calcul des Variations, 1861 p. 204; etc.
Take the equation of the catenary which was given in the preceding Chapter, Art. 30 in the form
It follows at once that
On the right-hand side of the equation stands a one-valued function, but on the left-hand side, a two-valued function. It is therefore necessary to define the left-hand side so that it will be a one-valued function corresponding to the right-hand side.
If we make , then is
and consequently is positive. But when , it is seen that
and then is negative. It therefore follows that there is only one root of , and this is for the value . The corresponding value of is .
This value is the smallest value that can have; for is the condition for a maximum or a minimum value and since is positive for , it follows that is a minimum value of . Further, since is continuously positive or continuously negative, there is no maximum value of . The tangent to the curve at the point , is parallel to the -axis, since at this point .
At every point of the curve we have
Hence, to construct a tangent at any point of the catenary, for example at , drop the perpendicular , and describe the semi-circle on as diameter. Then, with radius equal to , draw a circle from as center, which cuts the semi-circle at ; join and . The line is the required tangent.
where denotes that the arc is measured from the lowest point of the catenary.
The geometrical locus of is a curve which cuts all the tangents to the catenary at right angles, and is therefore the orthogonal trajectory of this system of tangents. This trajectory has the remarkable property that the perpendiculars , etc., of length , which are employed in the construction of the tangents to the catenary, are themselves tangent to the trajectory.
This trajectory possesses also the remarkable property that, if we rotate it around the -axis, the surface of rotation has a constant curvature,
Further, , the normal to the catenary,
where is the length of the radius of curvature.
Article 36. The geometrical construction of the catenary. Take an ordinate equal to . This determines the point (see figure). With as center and radius equal to , describe a circle. This intersects at a point , say. On the circumference of this circle take a point , very near , and draw the line , and on this line extended take such that . With radius draw another circle, and on this circle take a point , very near the point , and draw the line . Take on this line extended the point , so that , etc. The locus of the points is the required catenary.
The accompanying figure shows approximately the relative positions of the catenary, its evolute and the trajectory.
It appears trom the previous article that a catenary is completely determined when we know any point on it and the tangent at this point. This may be proved analytically as follows:
Let , be a point through which passes a straight line, making with the -axis an angle whose tangent is . The conditions that a catenary pass through this point and have the given line as tangent are:
For brevity write , so that the above conditions become
We therefore have
Since and are both positive, it follows that we may take only the upper sign. Consequently, if we write
Further, since has one and only one real value for a definite value of , the constant is determined uniquely from
and the quantities and determine uniquely a catenary which has the given line as tangent at the point ,.
In particular, consider the catenary that has the K-axis as the -axis of symmetry, and let the two points and be at equal heights on the curve so that their coordinates are, say and .
The equation of the catenary is now, since ,
say, where we regard as constant and variable.
We wish to determine whether this last equation gives a real value or real values for . We see that is infinite when and also when .
so that is negative infinity when is zero; is unity when is infinite, and changes sign once and only once as passes from zero to infinity. The least value that can have is for the value of that satisfies .
If, then, the given value of is greater than the least value of , there are two values of which satisfy ; if the given value of be equal to the least value of , there is only one value of ; and if the given value of is less than the least value of , there is no possible value of .
Moigno and Lindelöf have shown that the value of which satisfies
is approximately ; and then from  it follows that ; and therefore approximately (see Todhunter, loc. cit.. Art. 60). Thus there are two catenaries satisfying the prescribed conditions, or one or none according as is greater than, equal to, or less than 1.50888...
If we write , it is seen that and are the two tangents to the catenary that may be drawn through the origin.
As the ratio is independent of it also follows that all the catenaries of the form , which may be derived by varying , have the same two tangent lines through the origin, the points of contact being and .
Returning to the catenary , we shall see that also here there are three cases which come under investigation according as:
I. Two catenaries may be drawn through the fixed points;
II. One catenary may be drawn through these points;
III. No catenary may be drawn through the two points.
We may assume that , , we would only have to change the direction of the -axis which we name positive and negative; or we might consider the case of and , where is the image of ; that is, the point symmetrically situated to on the other side of the -ordinate.
From the equation of the catenary it follows that
and from this relation it is seen that has a positive or negative sign according as . Hence, also,
Under the assumption that , we must first show that such a figure as the one which follows cannot exist in the present discussion. We know that
That is necessarily positive is seen from the fact that the ordinate corresponds to the value , and is a minimum. (See Art. 34.) Suppose that . By hypothesis , and further , and consequently . The form of the curve is then that given in the figure; and we have within the interval to a value of , for which the ordinate is greater than it is at the end-points. must therefore have within this interval a maximum value. But we have shown (Art. 34) that there is no maximum value of y;
and there cannot be the minus sign as in equation [I]; hence,
Eliminate from [a] and [b] and noting that in [a] there is the sign, we have two different functions of , which may be written:
two functions of a transcendental nature, which we have now to consider. We must see whether , have roots with regard to ; that is whether it is possible to give to positive real values, so that the equations , will be satisfied. If it is possible thus to determine , we must then see whether the values which may be derived from equations [a] and [b] are one-valued.
The first derivative is
On the right-hand side of this expression is positive, also is positive, and
is positive, if .
Hence is positive in the interval .
It is further seen that continuously increases within the interval , so that is the least value that can take.
I , has no root;
II. , has one root, ;
III. , has a root, .
, is outside of the catenary;
, is on the catenary;
, is within the catenary.
This may be shown as follows:
since when , ; and, therefore, when , . We also have
where the positive sign is to be taken, when , and the negative sign, when .
We also have . Comparing this equation with equation [II] above, and noticing the figure, it is seen that, when
, then is on the catenary,
, then is outside the catenary,
, then is within the catenary.
Hence, when , there is one and only one real root in the interval , and we can draw through the points and a catenary, for which the abscissa of the lowest point is .
Article 44. The discussion of. We saw (Art. 42) that
When changes from to , the quantity continuously decreases, and consequently becomes greater and greater. Hence if the expression takes the value , it takes it only once in the interval from to . That this expression does take the value within this interval is seen from the fact that, for , , where , so that has a negative value; but, for , , so that the expression must take the value zero between these two values of .
Let be this value of which satisfies the equation, so that
which is an algebraical equation of the eight degree in , or an algebraical equation of the fourth degree in .
Article 45. An approximate geometrical construction for the root that lies between and . In the figure it is seen that the triangles and are similar, as are also the triangles and ; hence, if is the length of the line , we have
By taking equal lengths on the two semi-circles and prolonging and until they intersect, we have as the locus of the intersections a certain curve. This curve must intersect the -axis in a point , say. Noting that
it follows that
which, compared with the equation [A] above, shows that
Article 46. Graphical representation of the functions and . The lengths are measured on the -axis. Equation [c] gives ; that is, the tangent to the curve at the point is parallel to the axis of . Further, , so that the negative half of the axis of is asymptotic to the curve . The branch of the curve is here algebraic, since , for , is algebraically infinite.
Article 47. Consider next the cvrve. It is seen that ; and also , so that the tangent at this point is also parallel to the axis of the . Further, the negative half of the axis of the is an asymptote to the curve; but the branch of the curve is transcendental at the point ; because logarithms enter in the development of this function in the neighborhood of , as may be seen as follows:
where denotes a power series in positive and integral ascending powers of hence; the function behaves in the neighborhood of as a logarithm.
We saw that
For the value the expression within the brackets is zero, and when , this expression becomes , and is negative. As seen above in the interval to , the expression
becomes greater and greater, so that between the value and , it is negative.
Furthermore, is positive between and , and negative between and .
Hence increases between and , and decreases between and ; and consequently is a maximum.
We must consider the function when is given different values and see how many catenaries may be laid between the points and .
Case I. .
In this case is nowhere zero, and there is no root of which we can use. There is also no root of , since and , so that , and there is no root (see Art. 43).
Case II. .
All values of other than cause to be negative, so that there is a root and only one root of the equation , and consequently only one catenary. In this case can never be zero; since , and , so that , with the result similar to that in Case I.
Case III. .
We have here two catenaries. One root lies between and , and often another between and , as is seen from what follows:
Since continuously increases in the interval , it can take the value only once within this interval.
In the interval , continuously decreases, so that if , there is no root of within this interval; but if , then there is one and only one root within this interval, and in the latter case there are two catenaries.
We must next consider the roots of . When , then is , so that there is no root of . But when , then ; and has the root , which was just considered.
A) When , has two roots; and when , has a root in addition to the root which belongs to .
B) But when , then there is only one root for , which lies between ; this root is denoted by .
From the formula (Art. 42) for and we have:
We consider the values of within the interval ; for , ; and for , . Consequently within this interval is positive, and therefore also ; and since , it follows that .
On the other hand, ; and since , we have . Moreover, within the interval , continuously increases, and , so that within the interval , has no root, and within the interval , one root.
Hence, under B), has a root , within the interval , and only one root, and has a root between and , and only one, making a total under the heading B) of two catenaries.
We have the following summary:
. no catenary;
. , one catenary;
. , two catenaries.
Article 51. On the consideration of the intersection of the tangents drawn to the catenary at the points and .
Case I. As shown above, ft there is no catenary, so that the consideration of the tangents is without interest.
Case II. .
Here the catenary enjoys the remarkable property that the tangents drawn at the points and intersect on the -axis. In order to show this, we must return to the construction of the tangents at the points and . It was seen (Art. 45) that points and were found on the semi-circumferences and such that ( in this case), and that then the lines and were the required tangents, which intersect on the -axis.
Case III. .
Then, as already shown, has two roots, one of which lies between and , and the other between and . Let these roots be and respectively. For the root , we have
We assert that here the intersection of the tangents at and lies on the other side of the -axis from the curve.
In order to show this we need only prove that
This is seen as follows:
Now, since within the interval is positive, and since lies within this interval, it follows that is positive. Therefore is negative, and consequently is negative.
REMARK. In this consideration the whole interpretation depends upon the fact that the root lies in the interval , and the same discussion is applicable to Case B), where , and where the root lies between .
Article 52. On the consideration of the root .
. When .
The root lies within the interval and here is negative within the interval; therefore is positive, and consequently
so that is on the same side of the -axis as the curve.
. When ; then the root is a root of the equation , so we have here to consider the sign of
within the interval .
We have proved that within this interval is positive, and since
is positive, it follows that
is negative. Hence
Since is a positive quantity, it follows a fortiori that
and the intersection lies on the same side of the -axis as the curve.
We have seen that two catenaries having the same directrix cannot intersect in more than two points and . Denote as above the smaller parameter of these two curves by and the larger by . Then it is seen that , the curve of smaller parameter, comes up from below and crosses , the catenary of larger parameter, and, having crossed , never finds its way out again. For, consider the tangent to the curve as the point moves along this curve. This tangent must at first intersect , but at the vertex it is parallel to the -axis and evidently has no point in common with . Hence, for some position between these two positions the tangent to must also be tangent to see that there are two tangents common to and , and we shall next show that they intersect on the directrix.
Draw the common tangent and draw a tangent to the curve . Then between these lines we may lay an infinite number of catenaries that have the same directrix. One of these catenaries must be , for it touches and is the only catenary that can be drawn through the point of tangency made by (Art. 37). Consequently is the other common tangent to both curves.
We see also that the points and are beyond the points of contact of , with the two common tangents, while for the points of contact of the tangents are beyond and . It is also seen that, as the two curves and tend to coincide, the common tangents to the distinct curve become tangents to the single curve at the points and (see Art. 51). If we call the value of corresponding to this latter curve we have .
Suppose we have two catenaries which are not coincident and which have the same parameter . Denote their equations by
These catenaries intersect in only one point. For we have at once
which are the coordinates of one point.
Article 56. Lindelöf's Theorem (1860).
If we suppose the catenary to revolve around the -axis, as also the lines and , then the surface area generated by the revolution of the catenary is equal to the sum of the surface areas generated by the revolution of the two lines and about the -axis.
Suppose that with as center of similarity (Aehnlichkeits-punkt), the curve is subjected to a strain so that goes into the point , and into the point , the distance being very small and equal, say, to .
To abbreviate, let
denote the surface generated by ; that generated by ;
denote the surface generated by ; that generated by ; that by the catenary ; that by the catenary .
From the nature of the strain, the tangents and are tangents to the new curve at the points and , so that we may consider as a variation of the curve .
It is seen that
Now from the figure we have as the surface of rotation of
where denotes a variation of the second order.
a result which is correct to a differential of the first order.
In a similar manner
is an expression which is absolutely correct.
Article 57. Another proof.
We have seen that
and (see Fig. in Art. 45)
The surfaces of the two cones are, therefore, equal to
The surface generated by the catenary is
In the catenary (see Art. 35), so that
where we have taken the sign with because is negative, hence in [A] is negative.
But from 
Substituting in [B], we have, after making , for the area generated by the revolution of the catenary
which, as shown above, is the sum of the surface areas of the two cones.
Let us consider again the following figure, in which the strain is represented. In order to have a minimum surface of revolution, the curve which we rotate must satisfy the differential equation of the problem. If, then, we had a minimum, this would be brought about by the rotation of the catenary; for the catenary is the curve which satisfies the differential equation. But in our figure this curve can produce no minimal surface of revolution for two reasons: because, drawing tangents (in Art. 59 it is proved that there exists an infinite number) which intersect on the -axis, it is seen that the rotation of is the same as that of the two lines and , as shown above, so that there are an infinite number of lines that may be drawn between and which give the same surface of revolution as the catenary between these points; because between and lines may be drawn which, when caused to revolve about the -axis, would produce a smaller surface area than that produced by the revolution of the catenary. For the surface area generated by the revolution of is the same as that generated by . But the straight lines and do not satisfy the differential equation of the problem, since they are not catenaries. Hence the first variation along these lines is , so that between the points , and , curves may be drawn whose surface of rotation is smaller than that generated by the straight lines and .
The Case II, given above and known as the transition case, i.e., where the point of intersection of the tangents pass from one side to the other side of the -axis, affords also no minimal surface, since, as already seen, there are, by varying the quantity (Art. 56), an infinite number of surfaces of revolution that have the same area.
In Case III we had two roots of , which we called and , where . We consider first the catenary with parameter . This parameter satisfies the inequality
The equation of the tangent to the curve is
where and are the running coordinates. The intersection of this line with the -axis is
, or ;
Hence, when , , and when , .
On the other hand, is always positive, so that always increases when increases, and the tangent passes from along the -axis to , and never passes twice through the same point. It is thus seen that there are an infinite number of pairs of points on the catenary between the points and such that the tangents at any of these pairs of points intersect on the -axis, and there can consequently be no minimum. Such pairs of points are known as conjugate points.
When , the tangents intersect above the -axis, and there is in reality a minimum, as will be seen later.
Article 60. Application. Suppose we have two rings of equal size attached to the same axis which passes perpendicularly through their centers. If the rims of these rings are connected by a free film of liquid (soap solution), what form does the film take?
By a law in physics the film has a tendency to make its area as small as possible. Hence, only as a minimal surface will the film be in a state of equilibrium. Let be midway between and . The film is symmetric with respect to the and axes and has the form of a surface of revolution about the axis, this surface being a catenoid. The line is the axis of symmetry of the generating catenary. Construct the tangents and from the origin to the catenary. Only when and are situated beyond the rims of the circles will the generating arc of the catenary be free from conjugate points, and only then will we have a minimal surface and a position of stable equilibrium of the film.
We saw (Art. 38) that all catenaries having the same axis of symmetry and the same directrix may be laid between two lines inclined approximately at an angle to the directrix and which pass through the intersection of the directrix and the axis of symmetry. All catenaries under consideration then are ensconced within the lines and and have these lines as tangents. The arcs of these catenaries between their points of contact with and do not intersect one another. Through any point inside the angle will evidently pass one of these arcs, and the same arc (on account of the axis of symmetry of the catenary) will contain the point symmetrical to on the other side of . The arc contains no conjugate point (Chap. IX, Art. 128), and therefore generates a minimal surface of revolution. Further, this is the only arc of a catenary through the points and which generates a minimal surface.
Suppose that we started out with our two rings in contact and shoved them along the axis at the same rate and in opposite directions from the point . As long as and are situated within the angle (or what is the same thing, as long as ) then the tangents at and meet on the upper side of the -axis and there exists an arc of a catenary which gives a minimal surface of revolution and the film has a tendency to take a definite position and hold itself there. But as soon as the angle becomes equal to or greater than this tendency ceases and the equilibrium of the film becomes unstable. As a matter of fact (see Art. 101), the only minimum which now exists is that given by the surface of the two rings, the film having broken and gone into this form.
↑Throughout this discussion the -axis is taken as the directrix.
↑In other words, cannot be greater than and at the same time greater than .
↑The distance is, of course, measured on the -axis.
↑See also Todhunter, Researches in the Calculus of Variations, p. 29.