SECT. 3. Functions of Two or more Variables. Let u be a function of two variables x and y; then the conditions for u being a maximum or minimum, are In addition to these Lagrange has shewn* that the con must be impossible: from this condition it from this condition it appears that d'u da2 d2 u and must have the same sign; and the function dy2 will be a maximum if that sign be negative, and a minimum if it be positive. If the values of a and y which make = 0, du du 0, also dx dy make the second differentials vanish, there will be no maximum or minimum unless the third differentials also vanish, while the values of y' deduced from the biquadratic We may proceed in the same way to the consideration of cases in which the first total differential coefficient of u which does not vanish is of a higher order than the fourth. If u be a function of three variables wys, the conditions for a maximum or minimum are In like manner if there be a function of n variables x, y, ≈, t... we shall have, for determining their values when the function is a maximum or minimum, the n equations: In this case Lagrange's condition becomes too complicated to be easily expressed, and as such functions rarely if ever occur in practice, it is unnecessary to give it here. Français has shewn* that in a function of two variables, when the equations are satisfied simultaneously by the vanishing of one factor, they are really equivalent to only one condition, by which we cannot determine a and y, but can only find a relation between them. This corresponds geometrically to a locus of maxima and minima, such as would be produced by the extremity of the major axis of an ellipse which revolves round an axis parallel to the major axis. In these cases an equation which is usually excluded from Lagrange's condition. It is to be observed, however, that this view supposes a maximum to be a value not less than any other immediately contiguous, whereas it is generally considered to be a value greater than any other, and conversely for a minimum. This remark of Français is of more importance geometrically than analytically; and I may add, that in geometry the failure * Annales de Gergonne, Vol. III. p. 132. of Lagrange's condition indicates that there is a maximum for some sections, and a minimum for others. When a function of two or more variables is to be made a maximum or minimum, it frequently happens that there are given certain equations of condition between the variables, so that the real number of independent variables is less than the number of variables in the function. When this is the case, instead of getting rid of the superfluous variables by direct elimination, it is usually more convenient to employ Lagrange's method of indeterminate multiplierst. The following is the theory of the method. Let urbe a function of n variables 1, 2,..., these being subject to the r equations of condition, L1 = 0, L2 = 0,......L, = 0. When u is a maximum or minimum, which is to be combined with the r equations dL1 = 0, dL2 = 0, ... dL, = 0 (2), the general form of each of which is These equations are all linear in da, da,...dx,; mul tiply therefore the equations (2) by indeterminate multipliers A1, Ag...λ,, and add them to (1). We then get du + λ1d L1 + λgd L2 + ... + λ,dL, = 0; an equation which is of the form M1da1+ M2dx2 + ... + M2dx1 = 0, in which each quantity M is of the form If we determine the quantities λ from the conditions that they make the terms involving da1, dx2, ... da, vanish, that is to say, if we determine them by the conditions M1 = 0, M2 = 0, ... M1 = 0, + Mécanique Analytique, Vol. 1. p. 74. the variations dæ, dx2, ... da, are eliminated, and there remains Mr+dx++ ... + M1dx = 0: and as the nr quantities da,+1...da, are independent, their coefficients are separately equal to zero. mine A1, A2, ... A, by the equations Hence if we deter and then quantities ~ and the r quantities à satisfy the n+r equations, M1 = 0, M2 As it is indifferent which of the variables we eliminate in order to determine A, ... A,, the most general way of stating the result is, that we have the n+r equations, to determine the nr quantities X1, X2,..... X„, A1, A2, ..... Ar. If u be homogeneous in x1, x2, ..., and if L1, L2, ... L, consist of homogeneous terms and constants, there exists a simple relation between the quantities A, which is very useful in many problems. Let u be homogeneous of m dimensions, and let L1 = 0, L20, &c. be put under the form Ma+A=0, N2+ B = 0, &c. where Ma is homogeneous of a dimensions, N, homogeneous of b dimensions, &c., and where A, B, &c. are constants. Then multiplying equations (3) by x1, x2, ... x, and adding, we have, by a property of homogeneous functions, mu+aλ¿Ma + bλ2 N2 + &c. = 0, or b Therefore x = 0, or a = a, and y Lagrange's condition becomes 36xy - 9a2 > 0. Therefore a = 0, y = 0 gives neither a maximum nor a minimum, and x = a, y = a gives u = - a3, a minimum when a is positive, and a maximum when a is negative. x3 {(x2 - 1)3 -1} = 0, whence x = 0, x = ±2, y = 0, y x = 0, y = 0 give u = 0, a maximum; = = 2: x = 0, y (4) y = u = x3 y3 (a − x − y), a == give u = a maximum, 3 432 0, give u = 0 neither a maximum nor a minimum. xyz u = (a + x) (x + y) (y + ≈) (≈ + b) Taking the logarithmic differential, since logu is a maximum when u is so. Then 1 du 1 u dx = = 0, x x − y2 = 0, by − x2 = 0; |