then if n is replaced by z, the right-hand member of (137) vanishes by reason of (115), Art. 352; and therefore

is an exact differential; and consequently as ʼn or dy is arbitrary, zdнdæ is also an exact differential; and its integral is, by reason of (136),

d

B1 de (2)

dx

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 d'y vanishes at both limits. Then, replacing B, by its value given in (132), (138) becomes

Therefore the definite integral ['vde will be a maximum or a

d2v

minimum according as is negative or positive; provided dy'2 that it does not change sign nor become infinite between the assigned limits; and provided also that the constants c1 and c2 and are not such as to make zn'-nz' vanish or become infinite. It is worth remarking, that if n-n= 0, then 7dy;

in which case dн = O, 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

which is always positive, if the radical in v is affected with a positive sign.

Also, since the complete integral of H=0 is, see Ex. 2, Art. 351, y = c1 x + C2 i

c, and c, therefore must not be so assumed as to make c1+C2

C

= 0 for any value of x between the assigned limits.

358.] For a second example of the criteria, let

Let C1, C2, C3, C1 be the four arbitrary constants which enter into the complete integral of H = 0; then the value of z, which, substituted for n, satisfies (139), is

dy

c. ( de ) + c (dy ) + c (dy ) + c. (d); (140)

z = c12 (dy)

C3

so that, as before, zdн dx is an exact differential; and its inte

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

21

22

ž1 = c'2 (dy ) + c'2 ( d ) + c', (dy ) + c ́s (dy),

dc1

dc3

4dc4

where c'1, C2, C3, c'4 are other new arbitrary constants employed like the former ones in (140) to represent arbitrary variations of the constants c1, C2, C3, C4: so that z1 is a value of η which satisfies dн = 0.

it appears that any value of n, which makes dн = 0, will also satisfy the right-hand member of the equation; but dн = 0, if

whence, integrating by parts, and omitting the integrated part

which vanishes at the limits, we have

where, by virtue of (132),

c. = 1, (~2.) * = (--'—- *'-')* (d).

=

22

C2

=

And therefore for a maximum or minimum value of the definite

integral it is requisite that

d2v

dy") 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,

2-1 be n values of by expressed in the preceding forms, and containing n different series of arbitrary constants then the second variation is

of which the integral becomes, by neglecting the quantities at

d2v

wherein J, involves (), and another factor which is of the

form of a complete square; and where

and so on. It appears therefore that the maximum and minimum

d2v

value will depend on the sign of ((); and that it is neces

(n)2

sary that this latter quantity should not change its sign for any value of the variables between the given limits; and the arbitrary 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 :

Also, since the curve is determined by the differential equation,

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.