Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be [theta], so that OP = a sec[theta], and PB = b - a tan [theta]. If his rate of rowing is V miles an hour his time will be a sec [theta]/V + (b - a tan [theta]) sin [alpha]/V hours. Call this T. Then to the first power of [delta][theta], [delta]T = (a/V) sec²[theta] (sin [theta] - sin [alpha])[delta][theta], so that if AOB > [alpha], [delta]T and [delta][theta] have opposite signs from [theta] = 0 to [theta] = [alpha], and the same signs from [theta] = [alpha] to [theta] = AOB. So that when AOB is > [alpha], T decreases from [theta] = 0 to [theta] = [alpha], and then increases, so that he should land at a point distant a tan [alpha] from A, unless a tan [alpha] > b. When this is the case, [delta]T and [delta][theta] have opposite signs throughout the whole range of [theta], so that T decreases as [theta] increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not. The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values. In the case of a function of several variables, X1, x2, ... x_n, not independent but connected by m functional relations u1 = 0, u2 = 0, ..., u_m = 0, we might proceed to eliminate m of the variables; but Lagrange's "Method of undetermined Multipliers" is more elegant and generally more useful. We have [delta]u1 = 0, [delta]u2 = 0, ..., [delta]u_m = 0. Consider instead of [delta]u, what is the same thing, viz., [delta]u + [lambda]1[delta]u1 + [lambda]2[delta]u2 + ... + [lambda]_m[delta]u_m, where [lambda]1, [lambda]2, ... [lambda]_m, are arbitrary multipliers. The terms of the first order in this expression are __ [delta]u __ [delta]u1 __ [delta]u_m \ --------- [delta]x1 + [lambda]1 \ --------- [delta]x1 + ... + [lambda]_m \ ---------- [delta]x1. /__ [delta]x1 /__ [delta]x1 /__ [delta]x1 We can choose [lambda]1, ... [lambda]_m, to make the coefficients of [delta]x1, [delta]x2, ... [delta]x_m, vanish, and the remaining [delta]x_(m+1) to [delta]x_n may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put [delta]u [delta]u1 [delta]u_m ---------- + [lambda]1 ---------- + ... + [lambda]_m ---------- = 0 [delta]x_r [delta]x_r [delta]x_r