Then by the ordinary formula for the transformation of multiple integrals we get ...dev dxdx, ... dx = up” sin 0, sin ... sin dp de ̧do... do and the second number of the equation (13) by substitution will become [dpde, de ... 19 ... depsin V' p*1 sin 0,TMa sin 0,′′*...... sin 0,.,. (1 — n) ▼′... (14). n+1 (1 + p3) = ... ... But since u' is evanescent, we shall have p infinite, whenever æ ̧, ¤............∞, differ sensibly from x′′, x‚”‚.......x,”; and as moreover n-s+1 is positive, it is easy to perceive that we may neglect all the parts of the last integral for which these differences are sensible. Hence V' may be replaced with the constant value V in which we have generally = x = x.". Again, because the integrals in (14) ought to be taken from 0,0 to 02π, and afterwards from 0,0 to 0,π, whatever whole number less than 8-1 may be represented by r, we easily obtain by means of the well known function Gamma: fsin sin sin e ̧...... sin 0, de, do........ do... 8 and as by the aid of the same function we readily get and thus the equation (13) will take the form {(x, − x,'"')2 + (x, − x, '"')2 + ... + (x − x',')3 + u'3} In this equation V is supposed to be such a function of a xa, and u, that the equation (2) and condition (9) are both satisfied. Moreover V" 0, and V' is the particular value of V for which x = x," ; x1 = x," ;......x, x", and u = 0. Let us now make, for abridgment, = dV 0)..... P=u" (when u = 0). ..(b), and afterwards change x into x', and x" into x in the expression immediately preceding, there will then result dxdx......dx' P {(x,' — x,)2 + (x,' — x,)2 +...+ (x,' — x,)2 + u'3}' г 2 P being what P becomes by changing generally, into x,', the unit attached to the foot of P' indicating, as before, that the multiple integral comprises only the values admitted by the condition (a), and V' being what V becomes when we make u = 0. The equation just given supposes evanescent; but if we were to replace u' with the general value u in the first member, and make a corresponding change in the second by replacing ' with the general value V, this equation would still be correct, and we should thus have dxdx.......dr. P {x,' — x,)2 + (x,' — x2)2 + ... + (x,' − x ̧‚)' + u3} V............. ...(16). For under the present form both its members evidently satisfy the equation (2), the condition (9), and give V"= 0. Moreover, when the condition (3) is satisfied, the same members are equal in consequence of (15). Hence by what has before been proved (No. 4), they are necessarily equal in general. By comparing the equation (16) with the formula (1), it will become evident, that whenever we can by any means obtain a value of V satisfying the foregoing conditions, we shall always be able to assign a value of p' which substituted in (1) shall reproduce this value of V. In fact, by omitting the unit at the foot of P', which only serves to indicate the limits of the integral, we readily see that the required value of p' is 7. The foregoing results being obtained, it will now be convenient to introduce other independent variables in the place of the original ones, such that x1 =α11, x1 =α,1,... x, a11, u = hv, = a1, a,,... a, being functions of h, one of the new independent variables, determined by a‚2 = a,”2+h3‚ a; = a,'' + h3, ... a,' = a,2 + h2, and a function of the remaining new variables, 1, §2, §ã‚ ..... §. satisfying the equation a', a, a,... a being the same constant quantities as in the Equation (a), No. 1. Moreover, a, a,,... a, will take the values belonging to the extreme limit before marked with two accents, by simply assigning to h an infinite value. The easiest way of transforming the equation (2) will be to remark, that it is the general one which presents itself when we apply the Calculus of Variations to the quantity (5), in order to render it a minimum. We have therefore in the first place and by the ordinary formula for the transformation of multiple the expression (5) after substitution will become fde d... d dh aa ̧a ̧....... a ̧h* ̄*v* ̄ ̄ ̄1.... 2 A = or 2 a 29 a, Applying now the method of integration by parts to the variation of this quantity, by reduction, we get for the equivalent of (2) the equation where the finite integrals are all supposed taken from r 1 to rs+1, and from 1 to r's+1. The last equation may be put under the abridged form, and therefore the condition (9) in like manner will become απ' αξι dh (9'); s+1 where the values of the variables,,,,.... must be such as satisfy the equation = 0, whatever h may be; and as n is positive, it is clear that this condition will always be satisfied, provided the partial differentials of V relative to the new variables are all finite. 8. Let us now try whether it is possible to satisfy the equation (2") by means of a function of the form H depending on the variable h only, and & being a rational and entire function of E.,,,..., of the degree y, and quite independent of h. 3000 By substituting this value of Vin (2′′) and making |