and as du 0, and dx, dy, dz are arbitrary functions of x, y, z, we must have [a+B+y] = 0, == 0, н= 0, Y = 0: (25) (26) of which (25) is a series of equations in terms of the variables and their differentials at the limits, and the integrals of (26) will give the general functional relations between the variables: it is also to be observed that the same function will be given by any two of the three equations (26), for as ¤, н, v are all deduced by a similar process from a, the functional form of a will be, at least implicitly, contained in each; and therefore all the integral equations which may be deduced from them, provided that they have the same limits, will have the same functional form of this result many examples will occur in the sequel. In problems of critical values for which the Calculus of Variations is required, the discriminating criteria of maxima and minima are frequently as difficult of application as they are difficult of discovery; and I shall reserve the consideration of them to a future section. No practical inconvenience will be caused hereby, for I propose to investigate at present only those problems, the conditions of which will immediately indicate the nature of the critical values. 322.] To determine the shortest line joining, (1) two given points, (2) two given curves in the same plane. (1) Let (xo, yo) (x1,y1) be the given points; and let u = the length of the line joining them; then this is the equation to a straight line, which is therefore the shortest line; a, ß, a, b, being four arbitrary constants introduced in integration; these may be determined as follows: since the limits are fixed, dx = 0, dy = 0, dx1 = 0, dy1 =0; and therefore equation (27) is satisfied without any relation between the constants of the straight line and the limits: but as the line is to pass through the two points, x and y must satisfy simultaneously xo, Yo, and x1, y1; consequently (28) becomes which is the equation to a straight line passing through the two given points. (2) The process of determining the unknown function is the same in both parts of the problem; but in the second part the constants a, ẞ, a, b must be found as follows: from (27) we have then as dæ, dy are the variations of x and y as we pass from one point on the first limiting curve to another consecutive point, and as dæ1, dy1 are the similar quantities for the second limiting curve, and as (da), (dy); ds ds dx (dac), (d), ds ds are the direction-cosines of the straight line at the limiting points, (29) shew that the straight line cuts both limiting curves at right angles. Let (X, Yo), (X1, Y1) be the points where it meets F(x。, Y。) = 0, and F(1, 1) = 0 respectively, then we have also, as this straight line is normal to both curves, PRICE, VOL. II. 3 N thus these and the equations to the curve give four equations which are sufficient to determine the values of x, Yo, x1, Y1; and thus the problem is completely solved. 323.] Determine the form of the longest or shortest line which can be drawn from one curve to another curve in space. Let the equations to the curves be Thus a straight line, whose equations are (31), is the longest or shortest line joining two curves in space; and as dx dy dz are the direction-cosines of the line, and dro, dyo, dzo, dx, dy1, dz, are the variations of the limits, and consequently the projections of elements of the limiting curves, (30) shew that the straight line cuts both curves at right-angles; and from these conditions com bined with the equations to the limiting curves the unknown constants of (31) may be determined. The solution gives also the longest, as well as the shortest line, which can be drawn from one curve to another curve. The discriminating conditions of these two cases will be explained hereafter. 324.] Determine the critical value of uds, where ds is an element of a plane curve, and μ is a function of x and y. whence, squaring and adding, and substituting from equation (19), Art. 285, Vol. I, we have {(du) dy (du) where p is the radius of curvature at the point (x, y); and there dx dx ds dy ds S (34) This equation gives a geometrical property of the curve; but we cannot proceed further with the integration unless the form of μ is given. dr If the limiting values of x and y are given, the equations (32) are satisfied without any relation between xo, Yo, X1, Y1, ds 0 because dx = dy。= èx ̧= dy ̧=0; if the limits of integration are on two given plane curves, then (32) shew that the required curve cuts both the limiting curves at right angles. 325.] If ds in the last problem is an element of a curve in space, and μ is a function of x, y, z, then the equations of limits and of the indefinite terms become From (35) we infer, that if the curve is to be drawn between given limiting curves, it cuts both these curves at right-angles. |