A maximum, or greatest value, means one which is greater than any neighbouring value ; so that when a function is at its maximum, any allowable slight change must be one of diminution. For greater read leas, and for diminution increase, and we have the definition of a minimum. Now an ordinary question of maxima and minima is as follows :—sax being a function of x, what are the real values of x which make it a maximum or minimum i There is a maximum when x=a, provided that p and 1)(a —h), when both are possible, are both less than oa : but if one of the two ip (a h ) and 1) be impos sible, there is a maximum if both values of the other be leas than ipa. In all these cases it is supposes' that h may be as small as we please. Now 1. When ¢ (a+ h) and 1) (a—h) are both real, the theory explained in MAXIMA AND MINIMA is perfectly sufficient: there is a maximum when ¢'x changes from positive to negative in passing through ¢'a, and there is not a maximum in any other case.
2. When sr) (a + h) is impossible, there is a maximum if both values of (tix be positive from x= a —h up to xssa : when 1) is impossible, there is a maximum if both values of (p'x be negative from x=a to x=a+h.
It is the neglect of the second case which has led to the oversight in the calculus of variations which we shall presently mention. We shall now propose a case as follows :—It is required to find the maxi mum value of it in the equation The form of the curve which has this equation is as in this diagram ; o being the origin and o A (= A r) being unity. Now it ought certainly to be said that a P is the greatest ordinate of the curve, but neither is sa'x here equal to nothing, nor does it change sign. In fact when x = 1, we have sr) x = 1, (p'x =1.5. Tlio second criterion shows that A P is a maximum; the first shows uothing of the kind.
Now we can easily imagine it said, that in such a case as the preceding, A r, though unquestionably ( the greatest ordinate the curve can have, is not what is technically called a maximum : but it is meant that the last term should be restricted solely to denote those values of fx in which 1) (x + h) and l) (x-11) are both possible, and both less than cps.
To this, &kris paribus, there could be no objection : it often happens that the technical use made of a foreign term will not bear, and is not meant to bear, translation into our own language. The word ntaxi.uum, even in its widest allowable use, and if all we ask for should be granted, will not answer to greatest : for there may be several maxima and minima, and some of the minima may be greater than some of the maxima, which cannot be true of the words when translated. Suppose, then,
that the word maximum is so restricted as to apply to no value of /a except when sf) (x + li) and ¢ (x—h) are both possible: the disadvantage will be twofold. First, in every problem of maxima and minima, or in every problem which is reducible to one of maxima and minima, we shall have to invent an additional term to signify, perhaps, the very greatest or very least value of the function. Secondly, in applying the same limitation to tho calculus of variations we shall frequently be obliged to forego the solution of which we are in search, unless wo look for the very case, as an answer to a problem of maxima and minima, to which we have previously refused to apply the term maximum or minimum.
In order to as before described, a maximum, it is generally presumed that afvdx must= 0, and that y must be found in terms of x from this condition. Now the truth is thatfvdx, after the variation, becomes lvdx + bivdx, and all that is absolutely necessary is that b/vdx should be always negative, for all values of ax and by between tho limits, and for all values which are con sistent with the limiting conditions, at the limits. It is easily shown that this requires, as to the indefinite integral part, the follow ing equation :— r'+ o"—e+,... =0; and if we be resolved not to consider any points of the limiting curves, except those at which ax., 6x,, Sy„ may be either positive or negative, as we please, then it is easily proved that the rest of the expression for 6jvdx must also vanish, and this limitation is generally made in works on the subject, by which means solutions are misstated and may even be lost sight of. Thus it is generally asserted that the shortest line between two curves is always a straight line which is per pendicular to the tangents of both ; and that a flexible chain, allowed to elide between two curves, with an extremity on each, is in equili brium when it is in the form of a catenary perpendicular to the retaining curves at the points of suspension. On this we need only direct attention to the accompanying figure. The shortest line that can be drawn between the curves at P A and B (in is A 3, which is per pendicular to neither of the tangents A C, nn; and tho flexible chain A E 13 will hang from the cusps A and a without the slightest tendency to become perpendicular to A c and a D at its extremities.