one root of each pair, and call the three roots thus taken x, y, the other roots are therefore x, y, ૪. There d(-x)+ß 1-e2( − y)2 -ay-y3 The values of x and y are arbitrary; when they are given, a, B, and ≈ may be determined. 3 1 = &c. &c. Hence Ax 1 1 - e* y2 1 dx + dy + Ax ΔΥ and by the theory of equations x2y2x2 = ß2. It is immaterial which sign we ascribe to z. the upper sign, then yAx-x2 ▲y]2 = (y2 − x2) (1 − e2x2 y2). In accordance with what has been already said, the term in de of equation (a) will be to be taken negatively if the value we have assigned to ≈ does not make B▲ x + a≈ + x3 = 0, but on the contrary makes BA≈ - az 3 = 0, Now if we actually As being supposed always positive. express a≈ + 3 in terms of x and y, we shall find that it is always positive for values of x and y, which do not transgress certain limits. Hence the second of the last written equations must be taken, and therefore, if Ex denote we have, as x, y, z are zero together, Ex Ey - Ez = e2 xyz. ..... = (b). A little consideration will shew that as Ex-E(-x) the final result would be in effect the same if we had taken the lower sign in the equation Equation (b) is the fundamental equation for the comparison of elliptic arcs. Let ≈ 1: the corresponding values of a and y lie (e being less than unity) within the limits already mentioned. Now Ex represents generally an arc measured from the end of the minor axis, Ex will therefore be, when = 1, equal to the quadrantal arc of the ellipse, and consequently E (1) Ey will be an arc measured from the end of the major axis to the point whose abscissa is y, (y is of course not the ordinate corresponding to a). If the first arc is called s and the second s' or the difference between two elliptic arcs is expressed as an algebraical function of the corresponding abscissæ. This remarkable theorem was discovered by Fagnani*. As we have made = 1, we shall have the following relation between a and y, This relation admits of a simpler form, viz. 1 − x“ − y2 + e* x® y2 = 0. (6) The result at which we have just arrived admits of a simple independent proof, which is worth noticing, because it is easily remembered, and because it is, in effect, Fagnani's own demonstration of his theorem. As we know y2 ♦ Hence de-ds'-(-) dx + ("") "dy. = - y2 = 1 But in virtue of the relation in x and y, In fig. (63), let BMNA be a quadrant of the ellipse AC, the major axis being unity. Then if CP ber and CQ, y, we shall have Draw CY perpen This may be put in a different form. dicular to MY, the tangent at M and CZ perpendicular to NZ, the tangent at N: then we can shew that MY and NZ are equal, and that either is equal to the difference of the arcs BM and AN. For MY being the polar subtangent, is equal which is the form in which Fagnani's theorem is generally presented. That MY is equal to NZ is now evident, since the expression e2xy is symmetrical in x and y. Almost all the preceding results are included in Abel's theorem. This very remarkable theorem appeared in the third volume of Crelle's Journal; and though it is only a particular case of the general results which Abel communicated to the Institute in 1826, and which were published in the Mémoires des Savans Etrangers in 1841, yet in itself it is enough to place him in the first rank of analysts. The following demonstration of it is essentially the same as Abel's, but it is somewhat differently arranged. Taken in connection with the examples already given, it will not, we hope, be found difficult to follow. Lemma I. Let a be a root of the equation Fu=0, then, as we know, Consequently, if Пxu denote the term in in the ex и pansion of xu in a series of descending powers of u, χει This is true whatever integral value we assign to q; and therefore, if fr be an integral function of a, = [] fu Fu' Lemma II. fx Now consider the expression fx-fa x-a Dfx. Then is necessarily an integral function of a, call it is the (n + 1)th, if n be the degree of Fu, or there is no term |