Let C, and c, be the arbitrary constants contained in the integral of h = 0, and let then if n is replaced by 2, the right-hand member of (137) vanishes by reason of (115), Art. 352; and therefore is an exact differential; and consequently as n or dy is arbitrary, z8h dx is also an exact differential; and its integral is, by reason of (136), by me (2) Now suppose that n = dy = zo'y; .. ó'y = 1; where d'y is a new function of x; then, substituting in the second variation of the definite integral, we have And, to take the case most free from difficulty, let us suppose the limiting values to be fixed, so that dy and therefore o'y vanishes at both limits. Then, replacing B, by its value given in (132), (138) becomes gu = " (en(= mc) dr. Therefore the definite integral /'v dr will be a maximum or a d2v minimum according is negative or positive; provided that it does not change sign nor become infinite between the assigned limits; and provided also that the constants C, and c, and n are not such as to make zn' - nz vanish or become infinite. It is worth remarking, that if zn' -nz'= 0, then n=z=dy; in which case dH = 0, and therefore the second variation of the definite integral vanishes; and this is plainly inconsistent with the possibility of our deducing from it the criteria of maxima and minima. For an application of the preceding, let us consider the case of the longest or shortest line between two given points; here v = (1+y2); d2v 1 3. edit na atunci which is always positive, if the radical in v is affected with a positive sign. Also, since the complete integral of =0 is, see Ex. 2, Art. 351, y = C7X+C ; so that, as before, z8h dx is an exact differential; and its integral is, by virtue of (136), da ini (141) Lot where d'y represents a new variation of y; then, integrating the expression for the second variation of the definite integral, and assuming the limits to be fixed, so that the terms at them vanish, we have where c'1, c',, c'z, 64 are other new arbitrary constants employed like the former ones in (140) to represent arbitrary variations of the constants C, C2, C3, C4: so that 2, is a value of n which satisfies ôh = 0. Also since from (141) it appears that any value of n, which makes dh = 0, will also satisfy the right-hand member of the equation; but ÔH = 0, if n = 21, therefore (6) is a solution of the right-hand member: also (1) = 7777. Let 12 = () omy; and substituting this in (142) we have 8u =-"{B, va + di certos () "y dx ; whence, integrating by parts, and omitting the integrated part which vanishes at the limits, we have And therefore for a maximum or minimum value of the definite integral it is requisite that (2 ) should be respectively negative or positive for all values of the variables between the limits; also the second factor must neither vanish nor become infinite : the arbitrary constants therefore must be so determined as to fulfil these conditions. 359.) If the infinitesimal element-function of the definite integral contains derived-functions of y up to the nth, the process to be pursued is exactly similar to those of the two preceding particular cases; and therefore I need give no more than an outline of it. Let 2, 21, 22, ... 2n-1 be n values of dy expressed in the preceding forms, and containing n different series of arbitrary constants: then the second variation is d. An d2.a, n" 824 = 10T of which the integral becomes, by neglecting the quantities at the limits, reduced to mi 8u = d*-1.B,n,(n-1)) 3 B, nit +...+ neces 1 dév wherein J, involves 2), and another factor which is of the form of a complete square; and where zo'y = dy, () 'y = o'y, and so on. It appears therefore that the maximum and minimum value will depend on the sign of (20m2); and that it is necessary that this latter quantity should not change its sign for any value of the variables between the given limits; and the arbi. trary constants must not be such as to allow the other factor in (144) to vanish or to become infinite. 360.] We need not enter at length on the determination of criteria for relative maxima and minima, because we have shewn above that such cases are by means of an indeterminate multiplier reduced to those of absolute critical values, and the criteria determined for this latter case are therefore applicable to the former one. Let us however shew that the solution given in the fourth example of Art. 351 is a maximum : and (1 + y2) Also, since the curve is determined by the differential equation, H = 0 = y-ay); . _ d dy' and therefore the answer gives a maximum or minimum value according as y" is negative or positive. Let the origin be at the centre of the circle; then, since, as shewn by the value of u, the curve is taken in the first quadrant, y" is negative, and consequently the solution corresponds to a maximum. |