du

= 0 or ∞, or

du dy

= 0 or ∞. Let us however assume

dy

du

that is not indeterminate in form, and thus that

and

da

du

dy

are not simultaneously equal either to 0 or to ∞. du Suppose that

du

= 0, at a particular value for which

dy

does not vanish or become infinite, then the singular value day depends on the sign of dx2,

see Art. 150; being a maximum

d2y

or minimum according as is negative or positive; and by

dx2

hence if this expression is positive, there is a minimum, and if it is negative, there is a maximum value.

This method however of determining maxima and minima is

very incomplete, as it does not discuss the cases where (du), or any other of the partial derived-functions, becomes infinite; and it is therefore to be taken as a suggestion of the manner in which such problems are to be solved: the best plan is to determine the special maxima and minima values for each problem separately, as follows.

Ex. 1. It is required to find the maxima and minima values of y, having given, y3+x3-3axy = 0.

dy

dx

therefore there is a change of sign of from + to and

thus these values correspond to a maximum.

If = 0, it changes sign from to +, which indicates a

dy

dx

minimum.

SECTION 3.-Maxima and minima of an explicit function of two independent variables.

155.] Let u= F(x, y) be the explicit function of two independent variables x and y, of which the maxima and minima are to be determined.

Now observing the definition of maxima and minima, given in Art. 144, viz. that a maximum is greater, and a minimum less, than any and every value of the function in its immediate neighbourhood, it follows that if x and yo are specific values of x and y, which give such a singular value to F(x, y); then F (xo, yo) is greater or less, as the case may be, than any value corresponding to the variables, whether a infinitesimally varies while y does not vary, or whether y infinitesimally varies while a does not vary, or whether x and y are simultaneously and infinitesimally increased, or whether as a is infinitesimally increased y is similarly decreased, or vice versa. This property may be thus expressed: If F (xo, yo) is a maximum or a minimum, r(xoh, yo+k) is less or greater than F(xo, yo) whatever are the signs of h and k, which are infinitesimal increments, and in whatever manner the signs are combined; and also whether k = 0, when is increased or diminished by h; and whether h = 0, when y is increased or diminished by k.

156.] This being the definition of such singular values, I propose to extend to functions of two variables the theory of criteria which, in Art. 150, has been applied to those of one variable.

dr

nor dx

Let us in the first place assume that neither (d) vanishes for the values xo, yo; then, by Art. 140, (51),

dr

dy

xo + Oh

Yo+oki (6)

the meaning of the notation in the right-hand member of this equation being that x。 and y, are to be respectively replaced by xo+Oh and yo+Ok, where 0 is the general symbol of some positive and proper fraction. Now if F(xo, yo) is a maximum or a minimum, the left-hand member of (6) will have the same sign, h and k being infinitesimal, whether h and k are positive or negative, and when either one or the other of them is absolutely zero; and of course its equivalent must be subject to the same condition. But these circumstances cannot exist in the right-hand member of (6); that is, so long as the equivalent of the left-hand member takes the form given in (6). It is necessary therefore that it should have another form; and such it will take only when, for x = xo and y = yo,

If the left-hand member satisfies the second condition above stated for a maximum or a minimum, viz. that the sign of it is the same whether either h or k, but not both, is zero, it is necessary that

that (d) and (d) should be of the

dy2

same sign. Also

of the quadratic expression in the right-hand member the first and third terms are the same whether h and k are positive or negative: but the sign of the second term may be either positive or negative, according as h and k have the same or different signs; it is necessary therefore that the relation between the three terms should be such that the sign of the whole expression within the

brackets should not be altered by changes of sign of h and k. This condition will be obtained, if the roots of the expression are imaginary; because in that case it will be the arithmetical sum of two squares. The roots of a quadratic are imaginary, if four times the product of the first and last terms is greater than the square of the middle term: that is, in this case, if

If this condition is satisfied, the sign of the right-hand member

(d2x) and

F(xo, yo) is a maximum or a minimum according as

d2F

dy2

are negative or positive. Hence, if r (xo, yo) is a maximum

or a minimum, when x = x。 and У = Yo,

and F(xo, yo) is a maximum or a minimum according as

and (127)

d2F

dx2

d2F dy2 been first determined by Lagrange, is known by the name of Lagrange's condition.

are negative or positive. The inequality (9) having

Examples of illustration are given in a subsequent Article.

But it is clear that the right-hand member, being of uneven dimensions in terms of h and k, will change sign with h and k; and therefore if the equivalent of the left-hand member takes

(11)

the form given in (11), F(xo, yo) cannot be a maximum or a minimum. If therefore all the first and the second derived-functions vanish when x = x and y = yo, F(xo, yo) cannot be a maximum or a minimum unless also all the third derived-functions vanish; in which case the equivalent of the left-hand member of (6) involves fourth derived-functions, and the sum of the powers of h and k is four in every term. This circumstance is consistent with a singular value of r (xo, yo), if the biquadratic expression in terms of h and k is subject to conditions analogous to that given in (9), so that it should not change sign with h and k. The conditions however are such that it is unnecessary to state them. And the singular value will be a maximum or a minimum according as are negative or positive.

(dar)

and (d)

dy

And generally if F (xo, yo) is a singular value, all the partial derived-functions vanish for the particular values of the variables up to those of an odd order inclusive; and those of the even order, which are the first not to vanish, must be so related that the sign of the expression involving h and k should be the same, whatever are the signs of h and k.

158.] Let us however consider the subject of maxima and minima of functions of two variables from another point of view. If F(x, y) has such a singular value, it is necessary that it should exist for infinitesimal variations of a when У does not vary; and for infinitesimal variations of y when x does not vary. If the function fulfils the former of these conditions, we shall call it an x-partial maximum or minimum; and if it fulfils the latter a y-partial maximum or minimum. And when these partial maxima or minima are combined, we have what may be called a total maximum or minimum. The criteria of such a total maximum or minimum are now to be discussed. If however a partial maximum or minimum is combined with a partial minimum or maximum, the value does not fulfil the conditions of the singular values which are given in Art. 144, viz. that a maximum value is greater, and a minimum value is less, than each and every value of the function in its immediate neighbourhood. Let us express these conditions and circumstances mathematically. If F(x, y) is that function of which the x-partial singular value is to be found, it is necessary that (1) should

dx