termed absolute : then, by the theory of maxima and minima, it is plain, if u has a critical value, that du = 0 and changes its sign; and that the change of sign may be determined by the sign of 8u; so that if du = 0, u has a maximum or minimum value, according as ou is negative or positive; the solution of the problem therefore requires the calculation of ou and of 82u ; and by the condition du = 0, the form of the functional symbol connecting the variables is to be found. In the first place, let (1) where n = F(x, da, da, ... dra, y, du, đều, ... "y), (2) and F is the symbol of a known function. On referring to the value of ou given in equation (51) of the preceding Chapter, it will be observed that it consists of two parts; one of which is integrated, and depends on the values of the variables, of their differentials, and of their variations at the limits; the other is under signs of integration, and cannot be further reduced, because Ôx and oy are unknown functions of x and y, and because the other factors in the element-functions involve the undetermined function and its differentials. What conditions therefore are requisite so that ou = 0? For convenience of reference let {X-dx, +d2x,–...(-)"-12"-1x,} ôx +x,dn-1dx = a; (3) and let the analogous quantity involving ồy and its differentials = B: also let X-dx, + d’x,-...(-)"d"x, = 5, Y-dy,+dY, - ...(-)" do yn = h; so that we have ộu = [a+B] + "(38x + hey) Now, as ôx and oy are arbitrary functions of x and y, du cannot vanish unless [ a +87= 0; whence we have canna се (8) and also, I= 0, (9) H = 0; (10) and these are the conditions which are primarily necessary to the definite integral u having a maximum or a minimum value. 317.] Although it is desirable, both for symmetry and for the discussion of an expression in its most general form, thus far to retain all the terms in ou, and although in many of our subsequent examples we shall retain them throughout, yet it is necessary somewhat to abridge them, that we may point out some general properties of the above equations. First, let the difference between (7), (8), and (9), (10) be observed : (7) and (8) involve limiting values of ox, dy, and of their differentials; whereas E = 0, and h = 0, being differential expressions, will after integration give general relations between x and y, and therein the required functional connection; and the same function will be deduced both from 2 = 0 and from h = 0, provided that (and this is a necessary condition) the same limiting values are taken in the integrals of both equations : for the form of the function involved in them will depend on the form of function of n, and from a they are derived by a similar process; and therefore the same functional form will appear in the final result of each. Again, let us suppose that there is no variation of x, save at the limits; and that consequently the shifting of any point from a curve to the next consecutive curve is due to a variation of y only; then ôx = 0 (except at the limits), dòx = d28x = ... =0: so that (6) becomes each of the three lines of the second member of which must separately = 0, if ou = 0. Now, as we have shewn in the preceding Chapter, Art. 301, that the generality of the formulæ is not abridged by the assumption that one of the variables undergoes no variation, so the result hereby demonstrated is as general as that given in (7), ... (10). And as the solution of the problem is necessarily the same in both cases, we can infer that the same functional relation between x and y arises from s = 0, and from h = 0. 318.] Let us then take this last case to be the solution of the general problem of maxima and minima, and thereby indicate some general properties of it. And let us consider the case in which a is not linear with respect to dmy, so that you is a function of dmy, and consequently dmym involves d2my; thus h = 0 is a differential equation containing dumy; and as in process of integration a constant is manifestly introduced at each successive integration, so the complete integral involves 2m arbitrary constants; thus, if r is the complete integral, it involves C1, C2, C3,... Cam, that is, 2m unknown constants : and these must be determined by means of the former parts of equation (11), which are functions of the limits. Now if the limits are not restricted by any given conditions, the former parts of (11) will contain 2(m+1) arbitrary quantities, viz. Òxo, oyo, doyo, d28yo,...d"-1840, dx1, 841, d8y1, d28y1.... dm-1841, (12) of which the coefficients must be separately equated to zero : hereby we shall have 2(m+1) different and independent equations to determine 2m arbitrary constants, and which are manifestly more than sufficient, and thus the problem is indeterminate; this is as it should be ; for if there is no restriction on the limits or their variations, the definite integral may be of any magnitude, and cannot have either a maximum or a minimum value. If however equations are given connecting the variables at the limits; that is, if equations are given in terms of X, and yo, and in terms of x, and yı: then, if r = 0 is the integral of H = 0, there will be given dm-lgi Todyli (dyz )... (dym-i), Tu dylı *** dym-i); I which with the 2m + 2 quantities of (12) give us 4m +2 different quantities whereby to determine 2m constants C1, C2, ... Cam, and the 2m +2 quantities Xo, Yo, dyo, dạyo, ... dmyo, X1, Yı, dy,, d’yı, ... dmyii (14) and the problem is thus determinate. When .. is linear with respect to dmy, h = 0) will be a differ ential equation of an order not higher than 2m-1, and therefore its complete integral cannot contain more than 2m-1 arbitrary constants; and the number of equations relative to the limits of the general integral being the same as before, the problem is impossible, because the required conditions cannot be satisfied. 319.] The following are cases wherein the differential equation H = 0 takes particular forms, which admit of integration. (1) If n does not contain y, then h becomes -dy,+ d’Y,- ...(-)-1dmym = 0, (15) which admits of one integration without any determination of relation between y and X. (2) If a does not contain the first k terms of y, dy, dạy, ..., then h= 0 becomes (-)-1dky,(-)*dk+lYX+1–...(-)-1d"Ym = 0, (16) which admits of being integrated k times in succession. (3) If o does not contain x, or any differentials of x, then from (48), Art. 302, and from h = 0, we have dn = ydy + y, d.dy + y,d.dy+yzd.dBy + ..., 0=Y-dy, + dạyg-d’Y, +...; so that eliminating Y, da = Y, d.dy + dy dy, + Y d.day-dy dạy, + y2d.döy+dy døy, +... = d(y, dy)+d(y,dy-dy dyz) +d(Ygdøy- dyzdạy + dy d’yz) + ... whence by integration, B = (1 + Y, dy + y day-dy dy, +Yzdøy – dạy dyg+dy doyz +... (17) 320.] Thus far I have supposed the variables x and y, which are involved in the element n, to be independent of each other, and the maxima and minima of such definite integrals are called absolute. If however x and y are not independent, but are subject to a certain condition given by the equation, integral or differential as the case may be, L = 0, (18) then, as explained in Art. 307, we have OL = 0; (19) and a relation is given which the variations of the variables and their differentials must satisfy; multiplying therefore ol by an indeterminate constant multiplier », and adding to du, we have 8{u+AL} = 0; (20) and we may operate on utal in a manner precisely the same as that by which we have determined the necessary conditions for the critical values of u = n. These are called relative maxima and minima, and the method of determining them is hereby reduced to that of finding absolute maxima and minima. It is also similarly manifest that if the problem is the determination of the maximum or minimum value of u, when the variables and their differentials are subject to conditions expressed by a series of equations, which may be in the form of definite integrals or otherwise, viz., L =li, L2 = , ... L = las (21) then it is sufficient to determine the absolute critical value of u+1,42 +12 Lg + ... +1Lk (22) where 1,, d2, ...d are undetermined constants; and these will be determined by means of the necessary equations arising from equating to zero the variation of (22), and from the equations (21). I may also observe that the indeterminate multipliers , 12... 1, may be introduced in such a form that they may be supposed to be subject to variation, and thus to be functions of x and y. For suppose the function whose critical value is required to be u+1,(1,-1)+(12—12)+...+12(Lx-1); then the total variation of this quantity is du +1,04 +120 Lg + ... +1,01% +(1,-1,)81, +(12-12)d1, +...+(12-1)d1H and this, by reason of (21),=0, under the same conditions as those for which the variation of (22) = 0. 321.] The preceding principles are also applicable to the determination of the critical values of u where u = ["n, and a = F(x, dx, dx,...d"x, y, dy,...d"y, z, dz,...dkz); (23) and employing substitutions similar to those of Art. 316, |