follows the limits of the multiple integral (1) to be determined by the condition (o)*. In like manner it is clear that when $+£+~+$>1 • w. the expansion of V in powers of u will contain none bnt the even powers of this variable. Again, it is quite evident from the form of the function V that when any one of the » + l independent variables therein contained becomes infinite, this function will vanish of itself. 8. The three foregoing properties of V combined with the equation (2) will furnish some useful results. In fact, let us consider the quantity where the multiple integral comprises all the real values whether positive or negative of a^, xt,...x„ with all the real and positive values of u which satisfy the condition S+&'+-+S?+j?<1 '•«» Of, at,...a, and h being positive constant quantities; and such that we may have generally ar>ar'. In this case the multiple integral (5) will have two extreme limits, viz. one in which the conditions -^ + -^ + ... + -^ + Ti"*1 an^ "= * positive quantity... (7) * The necessity of this first property does not explicitly appear in what follows, but it roust be understood in order to place the application of the method of integration by parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when Y possesses this property, the theorems demonstrated in these Nos. ore certainly correct: but they are not necessarily so for every form of the function V, as will bo evident from what has been shewn in the third article of my Essay On 'he Application of Mathematical Ancdyiit to the Thtoriet of Electricity and Maonetiem. [See pp. »3-n] /^> Moreover, for greater distinctness, we shall mark the quantities belonging to the former with two accents, and those belonging to the latter with one only. Let us now suppose that V" is completely given, and likewise V[ or that portion of V in which the condition (3) is satisfied; then if we regard F,' or the rest of V as quite arbitrary, and afterwards endeavour to make the quantity (5) a minimum, we shall get in the usual way, by applying the Calculus of Variations, -lix,dx,...dxu"ir;^- (8), srcing that BV -0 and SF/=0, because the quantities V" and r/ are supposed given. The first line of the expression immediately preceding gives generally -V&S+Hr'Z «• which is identical with the equation (2) No. 1, and the second line gives JV 0 = «'""*-j-A («' being evanescent) (9). From the nature of the question de mini mo just resolved, there can be little doubt but that the equations (2') and (9) will suffice for the complete determination of V, where V" and V[ are both given. But as the truth of this will be of consequence in what follows, we will, before proceeding farther, give a demonstration of it; and the more willingly because it is simple and very general. 4. Now since in the expression {I, u is always positive, every oae of the dements of this expression will therefore he positive; and as moreover V" and Vt' are given, there must necessarily exist a function F0 which will render the quantity (5) t proper minimum. But it follows, from the principles of the Calculus of Variations, that this function F0, whatever it may he, must moreover satisfy the equations (2*) and (9). If then there exists any other function Vv which satiffies the last-named equations, and the given values of V" and F,', it is easy to perceive that the function will do so likewise, whatever the value of the arbitrary constant quantity A may he. Suppose therefore that A originally equal to zero is augmented successively hy the infinitely small increments SA, then the corresponding increment of V will he and the quantity (5) will remain constantly equal to its minimum value, however great A may become, seeing that hy what precedes the variation of this quantity must be equal to zero whatever the variation of V may he, provided the foregoing conditions are all satisfied. If then, besides V9 there exists another function F, satisfying them ell, we might give to the partial differentials of V, any values however great, hy augmenting the quantity A sufficiently, and thus cause the quantity (5) to exceed an} finite positive one, contrary to what has jnst been proved. Ilence no such value as F, exists. We thus see that when V" and Ft* are both given, there is one and only one way of satisfying simultaneously the partial differential equation (2), and the condition (9). 5. Again, it is clear that the condition (4) is satisfied for the whole of F,'; and it has before been observed fNo. 2) that when V is determined by the formula (1), it may always be expanded in a series of the form V~A + Eu,+ Cu■ + &c. Hence the right side of the equation (9) is a quantity of the order u'"~m; and u being evaneseent. this equation will then evidently be satisfied, provided we suppose, as we shall in what follows, that ,' n - a + 1 is positive. If now we could by any means determine the values of V" and VI belonging to the expression (1), the value of V would be had without integration by simply satisfying (2') and (9), as is evident from what precedes. But by supposing all the constant quantities at,a,,at a, and h infinite, It is clear that we shall have 0=P", and then we have only to find P,', and thence deduce the general value of V. 6. For this purpose let us consider the quantity dV_dU dVdU) (i dx, dx, da du)'' the limits of the mulciple integral being the same as those of the expression (5), and U being a function of xv xt,...... x, and u, satisfying the condition 0=>U" when av at, a, and h are infinite. But the method of integration by parts reduces the quantity (10) to since 0 = V"; and as wc have likewise 0 --*- U", the same quantity (10) may also be put under the form dV -Jdx^x, &c,-fa ^"-*- & Supposing therefore that U like V also satisfies the equation (2'), each of the expressions (11) and (12) will be reduced to its upper line, and we shall get by equating these two forms of the same quantity: Sdx1dat...dx,^- u"" V = fdxldx,...dx,^ u'-'U'; the quantities bearing an accent belonging, as was before explained, to one of the. extreme limits. Because V satisfies the condition (9), the equation immediately preceding may be written fdx1Jxt...dx,d-^u,*-,V, = fdxldxt...dx.*%u»-'U;. If now we give to the general function U the particular value u- {(xl-x;y+(xt-x;y+... + (x,-x;y+u>f?, which is admissible, since it satisfies for V to the equation (2), and gives U" =0, the last formula will become dV dxldxt...dx,u"-'-j l in which expression u' must be regarded as an evanescent positive quantity. In order now to effect the integrations indicated in the second member of this equation, let us make x, - a;," — up cos 6X; xt - xa" = up sin 0, cos 0,; xa — xt" = up sin 0, sin 0, cos 0„ &c. until we arrive at the two last, viz., x,_t — x',_x = up sin0,sin0,... sin 6,_tcos#,_,, u being, as before, a vanishing quantity. |