ON CERTAIN CASES OF GEOMETRICAL

MAXIMA AND MINIMA.*

WE occasionally meet in Geometry with certain cases of maxima or minima, for which the ordinary analytical process appears to fail, though from geometrical considerations it is obvious that maxima or minima do exist. The explanation of this failure is not given in works on the Differential Calculus, and some notice of it here may be acceptable to our readers. The difficulty and its explanation will be best seen in an example, and none is better suited for the purpose than a question proposed in one of the papers of the Smith's prize examination for 1842. This was-To explain the cause of the failure of the ordinary method of finding maxima and minima, when applied to the problem of finding the greatest or least perpendicular from the focus on the tangent to an ellipse, the perpendicular being expressed in terms of the radius vector.

The usual expression for the perpendicular in terms of the radius vector is

and as p2 will be a maximum or minimum when p is so, the ordinary rule for finding maxima and minima gives us

Now this equation can be satisfied only by r=+∞, values which are not admissible in this case whereas we know from geometry that p is a minimum when r=a (1 − e), and a maximum when ra (1+e).

It would appear then that these two values are not given by the analytical process, and the cause of this exception is to be explained. In the general theory of maxima and minima, it is assumed that the independent variable may receive all possible values; whereas in the present case r is limited to those values which are found by assigning all possible values to in the expression

a (1 − e2)

1- e cose

in other words, r is not absolutely independent. Now r being a function of another variable, admits itself of maximum and minimum values; and these are the values for which p is a maximum or minimum. The cause of the failure may therefore be thus exhibited: the equation

is satisfied by dr = 0, that is, by making r a maximum or minimum. Hence generally, if we wish to find the maximum and minimum values of y=f(x), we must consider, dy not only the values of x which satisfy the equation 0, dx but also the maximum and minimum values of x itself. In Liouville's Journal, Vol. VII., p. 163, there is given. a similar case of failure of the analytical process in the problem-To draw the shortest or longest line to a circle from a point without it. If we take the line passing through the centre of the circle, and the given point O as the axis of X, and call a the distance of the point from the centre, c the radius of the circle, x the coordinate of any point P in the circle measured from the centre, we shall have

OP2 = a2 + c2 - 2ax, a maximum or minimum;

from which the usual process would give

d

dx

(OP2)= - 2a =0,

a nugatory result. In this case x is a maximum or minimum, while OP is a minimum or maximum, and therefore the equation to be satisfied is

d.(OP2)

= —

which is satisfied by dx=0.

2adx = 0,

In this example the difficulty may be avoided by taking our coordinates generally, so that x shall not be a maximum or minimum when OP is so. We shall then have, calling a and b the coordinates of the centre of the circle, the other quantities as before,

OP2 = a+b+c* - 2b √(c2 - x) - 2ax = maximum; whence, by the usual process,

and

bx
√ (c2 — x2)

ас

x = ±

√(a2+b2)

giving the two values of x, which will solve the problem.

A very instructive example will be found in the problemTo find those conjugate diameters in an ellipse of which the sum is a maximum or a minimum. If r and be

conjugate diameters, a, b the axes, we have

any two

The former of these results gives the equal conjugate diameters, the sum of which is, as we know, a maximum. The latter result implies that both r and r, are maxima or minima, or that they are the principal axes, the sum of which is a minimum. By a different method we might have obtained the minimum instead of the maximum value of r+r, by the usual process for determining maxima and minima. For since r2+r,2=a2+b2 and rr, sin0= ab, 0 being the angle between the axes, we have

2

2

2

This is satisfied by cos 0-0

1

or 07, implying that r and r, are the principal axes. In this case the maximum value of r+r, is given by d0=0, since the equal conjugate diameters are those which make the greatest angle with each other.

ON THE SOLUTION OF CERTAIN

FUNCTIONAL EQUATIONS.*

IN the Cambridge Mathematical Journal, Vol. III., p. 92, Mr. Leslie Ellis pointed out what appeared to him to be the essential difference between Functional Equations and those which are usually met with in the various branches of analysis. His idea is, that these classes of equations are distinguished by the order in which the operations are performed, so that, whereas in our ordinary equations the known operations are performed on those which are unknown, in functional equations the converse is the case, the unknown operations being performed on those which are known. As this view appears to me to be not only correct, but of very great importance for the proper understanding of the higher departments of analysis, I shall endeavour in the following pages to enforce and illustrate it.

On the preceding theory it is easy to see why the solution of functional equations must involve difficulties of a higher order than that of equations of the other class. For if we consider an equation as a series of operations performed on a subject, the operations being known and the subject unknown, the solution of the equation involves the finding of the subject, which may be done theoretically by undoing the operations which have been performed on it; that is, by effecting on the second side the inverse of the

* Cambridge Mathematical Journal, Vol. III., p. 239.