change its sign at the corresponding values of x and y and

and according as

dr

(7)

should be equal to zero or to in

=8 we will at present omit :

at the critical value changes its sign

from + to or from to, so is the corresponding singular value a partial maximum or a partial minimum. Similarly for the y-partial singular value it is necessary that

dr

dy

should change its sign at the critical value, and therefore

that, omitting the infinite value, (d) = 0; and according as

dr
dy

dy

the change of sign of is from + to

so is the corresponding singular value a partial maximum or a partial minimum. To render our notions as free as possible from confusion, I will consider first the subject of maxima, and subsequently that of minima.

Now as the total maximum arises from the combination of two partial maxima, it is necessary that the conditions which they require should be simultaneously satisfied. And therefore at such a total maximum we must have

dr

(dr) = 0,

= 0. dy

(12)

And as no relation is given between dr and dy, and as these equations are simultaneously true, the quality of r (x, y), which each of them separately represents, is also true for all infinitesimal variations of the function at its singular value; and consequently whatever is the direction along which that variation takes place.

The sufficiency of (12) for all directions of variation of the function at the singular value may thus be shewn. Let x and be expressed in terms of new variables § and ŋ by means of the following equations, wherein the a's and b's are constants,

y

η

(13)

And thus if

(dr) and (d) vanish, (de) and (d) also vanish.

dx

dy

And therefore, inasmuch as the constants in (13) are undetermined, the first derived-functions vanish, whatever is the direction of the infinitesimal variation of the function: and therefore thus far the conditions of a total maximum are satisfied.

Let the critical values of x and y which satisfy (12) be x。 and

Yo; we have to examine the change of sign which () and (dr)

dy

undergo at these values. In certain cases this change may be conveniently determined by an examination of the critical factors themselves, according to the method explained in Articles 145 and 146. In the general case however the following process gives the required criteria.

Let us assume that for the critical values of x and y all the second derived-functions do not vanish, and let it be remembered that we are investigating the conditions for a total maxidr

mum. Then by Art. 149, if for the critical values () and dx

dy

vanish, and change sign from to, the following ratios are both negative, viz.

D

(dr)

(16)

the total differentials being taken in both numerators, so that we may investigate the change of sign of the first partial derived-functions for changes of not only the corresponding variable, but also of both the variables simultaneously; otherwise we might get from the critical values singular values in certain directions and not in all directions.

Now
d2 F dy

dx dx

In (17) and (18) dæ and dy are arbitrary; in fact, so far as we have proceeded, dr and dy in (17) need not be the same as the dr and dy in (18). Let us however suppose them to be so and for convenience of expression, let us substitute as follows:

To examine the signs of (17) and (18), let us assume ℗ to represent some quantity to which both may be equated; so that we have

A dx + B dy
B dx + c dy = 0 dy;

= 0 dx,

whence, eliminating dr and dy, we have the following quadratic

of which the two roots are to be negative, since (17) and (18) must be negative, so that r (xo, yo) may be a maximum; therefore all the coefficients must be positive, that is,

A and c must both be negative,

and AC-B2 must be positive.

(24)

(25)

Hence the necessary conditions that ro, yo should render F(x, y) a maximum are the following:

159.] Similarly to determine a total minimum, it is requisite that it should be a partial minimum of a partial minimum, and therefore, supposing that (2)

dx2

(d), (d) do not all

dy

following the process of the last Article, the two values of 0, and therefore the roots of (23), must be positive; whence it follows, that a and c must be both positive, and that AC - B2 must be positive; and therefore the conditions that ao, yo should render F(x, y) a minimum are, that

160.] As (23) has two roots, and thus gives two determinate values to 0, so by reason of (20) and (21) de and dy will bear a determinate ratio to each other; it may perhaps hence be inferred that the criteria of maxima and minina, which are thus determined, are applicable for variations of the original function in only two certain directions. This however is not the case for the criteria are true and sufficient whatever is the direction of variation of the original function which is due to the combined variations of the variables.

Let the variables x and y be replaced by new variables έ and 7, with which they are related by the equations (13), Art. 158; then from (15) we have

dx2

72

dε dn

dy

72

d§ dŋ

dn2

2

and therefore whatever is the sign of the left-hand member, the

same also is that of the right-hand member. If therefore the left-hand member is positive, and thus fulfils the condition neces

7526

sary for a singular value, the right-hand member is also positive, and similarly fulfils the required condition. Now in the equations (13), which connect x, y, §, and ŋ, the directions of § and ʼn relatively to those of x and y, or, to speak geometrically, the angles through which the coordinate axes are turned, depend on ɑ1, ɑ2, b1, b2; and the equivalence (31) is independent of these constants: hence if Lagrange's condition is satisfied for any system of two variables, it is also satisfied for every other system which is connected with it by two linear equations of the form (13), Art. 158; and therefore if Lagrange's condition is satisfied for the infinitesimal variations of the variables in any one direction, it is satisfied for their variations in all directions. It is also to be observed that if the right-hand member of

d2r

d2r

dn2

(31) is positive, then from (28) and (30) (d) and have the same sign as (d) and (d); and therefore if all the

dy2

conditions of a total maximum or minimum which are given in (26) and (27) are satisfied for one direction of the variation of the variables, they are also satisfied for all directions and for all the circumstances of such total singular values as exist at the critical values of the variables.

If all the second derived-functions of the original function vanish at the critical values, the preceding results come to nought as however I have alluded to this case in Art. 157, it is unnecessary to say more.

Some examples of total maxima and minima of functions of two variables are subjoined.

161.] Ex. 1. To determine whether any, and what, values of x and y render x2y+ xy2-axy a maximum or minimum. F(x, y) = x2y + xy2 — axy;

= 2xy + y2―ay = y(2x+y-a);

dr

dx