neither of the first three of which satisfy Lagrange's condition; d2F2 d2r have (d) and positive, and a2 dx2 dy2 which is a positive quantity; .. x2y+xy2-axy is a minimum, viz. Ex. 2. To find a point within a triangle for which, if lines are drawn to the angles, the sum of their squares is a minimum. Take A, one of the angular points, see fig. 20, of the triangle, for the origin; and let the base AB = a, and the coordinates to c be h, k; and let the coordinates of P be x, y, and the sum of the squares = F(X, Y). .'. F(x,y) = x2+ y2 + (a−x)2 + y2 + (h− x)2 + (k − y)2, and as in both cases the change of sign is from have a partial minimum of a partial minimum, and therefore the necessary conditions of a total minimum. Ex. 3. Inscribe the greatest triangle in a circle. Fig. 21. Let one of the angular points of the triangle be at a, the extremity of the diameter ACB; and let P and Q be the other angles; let AC = CB = a. .. PAB = 0, QAB = 4; therefore by a property of the circle, PA = 2a cos 0, area of triangle = F(0, 4) () = 2a2 cos = 2a2 cos 0 cos o sin (0+); {cos 0 cos (0+) — sin 0 sin (0 + $)}, = 2a2 cos & cos (20+p) ; critical values of 0 and 4 give a maximum: and as ✪ the triangle is equilateral. Hence the greatest triangle that can be inscribed in a circle is the equilateral one. d2r 162.] In the case in which (121) (121) – (177) dy2 dy )is nega tive, the last term of the quadratic in @ of equation (23) is negative, and therefore the two values of are of different signs; whence, by means of (17) and (18), it follows that one of the partial singular values is a maximum, and the other is a minimum: and therefore the conditions requisite for a total maximum or minimum are not fulfilled. d2F 2 dx dy dx2 dy2 then the last term of equa tion (23) = O, and therefore one value of is zero; and there fore either (17) or (18) = 0, and therefore either (1) or (+++) dy undergoes no variation; whereas then there is a partial maximum or minimum with respect to one of the variables, the other is such that corresponding to its variations the function is constant; hence we have a locus of such partial maxima or minima. These several conditions will be more clearly understood from the geometrical analogues of them in the theory of curved surfaces; which however it would be premature to explain in this place, and therefore we reserve them until they naturally arise in the course of the treatise. SECTION 4.-Maxima and minima of functions of three and more independent variables. 163.] Firstly, let us consider a function of three independent variables, x, y, z, of the form u = F(x, y, z). Extending the principles of Art. 158 and 159 to this more general case, it appears that a total maximum or minimum of a function of three variables must arise from the combination of three several partial maxima or minima with respect to the several variables. And also, as any two of the three variables may vary, while the remaining one does not vary, it appears that the conditions of such a combination of two partial maxima or minima must be fulfilled. Which conditions are, d2F d2F d2F therefore (127). (17), (123) must be of the same sign; and dx2 dy2 dz2 the singular value of F(x, y, z) is a maximum or minimum, according as they are negative or positive. There is also another relation between the several partial second derived-functions; dy (ar dy dy and Dz (dr) dz must be of the same sign, and be negative for a maximum value, and positive for a minimum; and d2r dy d2 F dz + dx dy dx + dx dz dx ( d2F dx d2F + + dy dx dy dy2 (duda) d2r ) d dy dz dy Ꭰ dr d2F dx d2r dy dz dx dz dz dy dz Now for convenience of expression, let d2F + + (36) dz2 and as (34), (35), (36) are to be of the same sign, let us employ a process of reasoning similar to that of Art. 158, and let us assume to be the symbol for some quantity which is the same in all; then the following system results : (A−0) dx + G dy + F dz = 0 G dr+ (B-0) dy + E dz= 0 Fdr + Edy + (c-0) dz = 0. whence, by cross-multiplication, (A-0) (B-0) (c-0) - E2 (A-0) - F2 (B-0) — G2 (c-0) (37) +2 EFG = 0; (38) the common Discriminating Cubic, as it is called, and which has three real roots; and, when expanded, becomes (3 — (A + B + C) 02 + (BC + CA + A B — E2 — F2 — G2) 0 -(ABC+2EFG-AE2-BF2-CG2) = 0. (39) Of this equation the three roots are to be of the same sign, and the result is a maximum or a minimum, according as they are negative or positive; therefore, besides the former conditions (32) and (33), the following expression must be negative for a maximum and positive for a minimum, viz. : ABC2EFG-AE2-BF2-CG2. (40) Hence, that a function of three variables may have a maximum or a minimum value, the critical values must satisfy 3 +3 +1 ( = 7) conditions, viz. three of equations (32), three of equations (33), and one of equation (40). 164.] Lastly, let us consider the general case; and let F (X1, X2, ... Xn) (41) be a function of n independent variables, of which the maxima and minima are to be determined; and, for convenience of notation, let of which it is to be observed, that (1, 2) = (2, 1), ... (n, 1) = (1, n), .... Now the singular value of (41) must arise from the combination of n similar partial singular values due to the separate variation of each of the n variables; and therefore we must have same sign; negative, that is, for a maximum, and positive for a minimum. If is the quantity to which each may be equated, then we shall have the following equations, {(1, 1) —0} dx1 + (1, 2) dx2 + ...... + (1, n) dxn = 0 (2, 1) dx1 + {(2, 2) −0} dx 2 + ...... + (2, n) dx1 = 0 ...... ·; (43) (n, 1) dx1 + (n, 2) dx2 + + {(n, n) −0} dxn = 0 whence, by the elimination of the n quantities, da, dx2, ... dxn, there will result an equation in @ of n dimensions, all the roots of which are to be of the same sign; and according as they are positive or negative, will the corresponding value of the function be a minimum or maximum. The number of conditions which are required to be fulfilled may thus be found: As the total maximum or minimum arises from the combination of the several partial singular values, all the conditions which they involve must separately be satisfied. Hence it is easy to see, that when the equation in has been formed, there will be involved, and to be satisfied, in the coefficients of 1 |