Also for values of x and y found from these equations, х must preserve an invariable sign, whatever be the signs and values of Dx and Dy. From this we deduce the same results as in the preceding article. 232. There is no theoretical difficulty in finding the maximum or minimum value of an implicit function of two independent variables, nor in finding the maximum or minimum value of a variable which is connected with any number of other variables by equations when the whole number of equations is two less than the whole number of variables. For example, suppose we have two equations 0............(1), f1(x, y, z, u) = 0, f(x, y, z, u) = 0.. involving four variables x, y, z, u, and we wish to find the maximum or minimum value of u. We may eliminate one of the three variables x, y, z between the two equations; suppose we eliminate z; then we obtain one equation connecting x, y, and u; from this we find u in terms of x and and proceed in the ordinary way to investigate the maximum or minimum value of u. Or if we wish to avoid the elimination we may adopt the following method; consider x and y as the independent variables and differentiate the given equations (1); thus y, du du find dz and , and dx dy' dx andy; then for a maximum or minimum value of u dz du and must be zero. Thus, more simply, dy da O and dx dy (0 in equations (2), and then dz dz eliminate and ; the two resulting equations combined dx dy with (1) will determine the values of x, y, z and u, which may correspond to a maximum or minimum value of u. And by differentiating equations (2) with respect to x and y we can find d'u d'u dx2, dx dy' d'u and and so settle whether u is really dy' a maximum or minimum. Practically the solution of problems of this class is facilitated by the method of indeterminate multipliers, which is explained in the following chapter. 233. The student will find it advantageous to illustrate this chapter by means of the Geometry of Three Dimensions. If z=6(x, y) be the equation to a surface, to find the maxima and minima values of z amounts to finding those points on the surface which are at a greater or less distance from the dz plane of (x, y) than adjacent points. The conditions = 0, dx dz and = 0, make the tangent plane at any one of the points dy in question parallel to the plane of (x, y). The interpretation of the case in which B2-AC-0 will be seen from what is stated in Art. 235. = The method given in Art. 231 admits of clear geometrical illustration. If, for example, there be a point on the given surface which is at a maximum distance from the plane of (x, y), then in passing from that point to an adjacent point, along any curve whatever lying on the surface, we must approach nearer to the plane of (x, y). Now, by combining the equation z=(x, y) with y=(x), we obtain a curve lying on the given surface, and by giving every variety of form to (x) we may obtain as many curves as we please. Hence we see that if we put y=(x), and leave the form of the function (x) arbitrary, we do not really break the restriction that x and y are to be independent. 234. A function u of two variables may have a maximum du or minimum value for values of x and y which render dx du and indeterminate or infinite. Such exceptional cases must dy be examined specially, as there is no general theory applicable to them. For example, suppose u = (x2 + y2)*, du du Here, when x and y vanish and become indeter dx dy minate. If we put y = ax, we have du du Hence and are infinite when x=0, and y = 0. But dx dy u is really a minimum then, for it vanishes only when x and y vanish and is never negative. 235. On a case of maxima or minima values of a function of two independent variables. х If u denote a function of two independent variables x and y, the values of x and y that make u a maximum or minimum are found from the two equations If these equations are satisfied by a single relation between x and y, we cannot determine a finite number of values of x and y, that render u a maximum or minimum. This case we propose to examine. Now suppose that from M=0, we find y in terms of x, say y=(x), and substitute in u; we thus make u a function of x only. On this hypothesis = dx' 0, since M=0 by hypothesis. Hence, this substitution of (x) for y has reduced u to a constant, since cular value to x. du dx vanishes without our assigning any parti- · Let us now return to equations (1) and (2). Change in (x, y) the variables x and y to x+h and y+k respectively. Calling u' the new value of u, we get Let us now assign to x and y any values consistent with (3), leaving however the ratio of k to h quite arbitrary, and examine whether u' becomes less or greater than u when k and h are sufficiently diminished. the above value of u', is ha The coefficient of in dx2+ h dx dy h2 dy3 or A+ 2 2k ka Now by (4) this kB\2 2 =A[1+ 9 and is therefore necessarily positive if A be positive, and necessarily negative if A be negative, whatever be the ratio of k to h, except for that particular value of the ratio which makes the coefficient vanish. Hence the conclusion will be this: if we assign to x and y values consistent with M= 0, then when h and k are sufficiently diminished, u' is certainly less than u d'u d'u if be negative, and certainly greater than u if be da dx2 positive, excepting only when k has to h one particular ratio. This latter case would require further examination, had we not already shewn that by a certain supposition u is reduced to a constant, so that when k has to h the one particular ratio, u' is ultimately neither greater than u nor less than u, but equal to it. The whole theory may be illustrated geometrically; for example, if z2 = a3 — x2 — y2 + (x cos a + y sin a).......... (1), find maxima or minima values of z; |