IV. To calculate by means of the potential of the magnetic fluid distributed over two spherical surfaces near AB at distances a on each side. If p, p' be the quantities of attracting and repelling magnetic fluids on an unit of each surface. the quantity in one of the poles of the small magnets, corresponding to an area A, since it is distributed over an arca X 2 for the surface radius r+e, and the potential with respect to a point at a distance x from the vertex, is dS', ds being corresponding elements of the surfaces whose radii are r+, r respectively. The whole potential V of the two fluids is 5. ON THE EVOLUTE OF CUBIC CURVES. то By HENRY M. JEFFERY, M.A. (Continued from p. 81). find the Boothian equation to the evolute of a non-singular spherical curve of the third class. Let this curve, as the plane curve in §4, be denoted by the equation a§3 +bn3 +c+6d§ŋ = 0. The equation to its evolute may be deduced from the threepoint equation of § 2 by substituting (a§3 +bn3 +c+ 6d§n) {b2c2§® + c2a3n® + a2b2 +.....} + (§2 + n2 + 1)2 (8d3 + abc) {6d (a§n2 + b§3n + c§3n2) + (2acn3 +2ab – bc§3) §2 + (2ab + 2bc§3 — acn3) n2 + (2bc§3 + 2acn3 — ab)} = 0. This is also the Boothian equation to the evolute of a nonsingular spherical curve of the third order (-ax-by+c+6dxy = 0), since the two curves are reciprocal. 6. To determine the Boothian equation to the evolute of a non-singular plane curve of the third order. Let its Cartesian equation be The equation to the evolute may be obtained, from §§ 2, 3, by the following conversion: (a§3+bn3) {b2c2§®+c*a*n® +a2b2—24bcd2§*n—24acd”§n* — 24abd2§n — 2n3 (a2bc + 16ad3) – 2§3 (ab3c + 16bd3) — 2§3n3 (abc2 + 16cd2) − 24§3n* (2d* + abcd)} + (§2 + n2)2 (8d3 + abc) {6d (a§n2 + b§3n) +(2acn3+2ab- be§3) §2 + (2ab+2bc§” — acn3) n2} = 0. VOL. XI. L 7. To find the equation to the evolute of a cusped cubic referred to three-point tangential coordinates. Let its tangential equation be a1 p3 + 3b,q2r=0, so that the reciprocal cubic is thus denoted: 4b1a3a3 — 9a ̧13cß3y=0. As in § 2 the equation to the evolute is the eliminant of three equations a1ƒ3 + 3b,g3h = 0, a‚ fp2+2bgqr+b ̧hq2 = 0, fP+gQ+hR=0, where P is written instead of a (ap-bq cos C-cr cos B). It is found to be (a, p3 +3b ̧q2r) (4b ̧P3 − 9a ̧ Q2R) = a ̧b ̧ (4pP−8qQ+rR) (4A2)2. 8. To find the equation to the evolute of a cusped cubic referred to trilinear coordinates. Let its equation be a,a3a2 + 3b,b*cẞ*y=0. The equation to the evolute is seen to be, from the permanence of equivalent forms, 3 (a ̧P3 +3b ̧ Q2R) (4b,p3 — 9a,q3r) = a,b ̧ (4pP−8qQ+rR) (4▲2)2. 3. If cusped cubics have the same cusp and point of inflexion, and if the tangents at those points intersect in the same point, the common tangents of all these cubics and their evolutes envelop the same parabola. See § 14, Cor. 9. In the Boothian system of coordinates, the evolutes of the cusped cubics (a,§3 + 3b,n2 = 0, a ̧x3+3b ̧y2=0) are thus denoted: (a ̧§3 + 3b ̧n3) §3 — a ̧ (§2 + n3)2 (§3 − 2n3) = 0, (46 ̧§3 +9a ̧n2) §3 — 4b, (§3 + n)2 (§2 — 2n3) = 0. In this particular case of the semi-cubical parabola, the equations to the evolute degenerates into one of the fourth degree; the enveloped parabola degenerates into two points, in which the common tangents meet the line at infinity, as is shewn in fig. 27; the line at infinity also passes through the points of inflexion of both curve and evolute.* 10. To find the equation to the evolute of a cubic curve of the third class with double tangents. This curve is also of the fourth order. Let its equation be a ̧ p3 + 3a ̧p2r+3b ̧q3r = 0. 3 The equation to its evolute may be written (a, p3 +3a ̧ p3r+3b„q3r) (12a ̧b ̧aP* + 24a ̧TMb„P2 Q2 + 12a ̧3 Qʻ 3 3 3 -4a,b,PR-36a,a,b,PQR +9a,b,Q*R*) = (4▲3)3 b ̧ {p (12a ̧à ̧ ̧Ð2 — 4a,b ̧PR — 12a,a,” Q3) 3 1 11. To find the equation to the evolute of a cubic curve of the third class with double tangents, referred to Boothian coordinates. We have to find the eliminant of the three equations (see §4) § (a ̧ƒa + 2a ̧ƒ) +2b ̧gn+a ̧ƒ2+b„g2 = 0, It is seen to be ƒ§+gn = 0. 3 3 (b ̧§2+a ̧n3) (a‚§n” + a ̧n2+b ̧§3) = 2a, (a ̧ — b ̧) §n', which may be also thus expressed (a ̧§3 +3a ̧§2 + 3b ̧n”) (a ̧n2+b ̧§2)” +a‚ ̧b ̧ (§2+n2)2 {b, ̧§ (2n* — §1) + a ̧§n2} = 0. 12. To determine the evolute of a cubic curve of the third order with double points, referred to Boothian coordinates. Let the Cartesian equation of the curve be Generally, if the equation to a curve of the nth class be ¿" + nan"-1 = 0, its evolute may be denoted by the equation (E" + nan"-1) E" = (Ea + n2)2 {E2− − 2¿a1-¤‚3 +...+ (− 1)”-1 (n − 2) §2‚2 + (− 1)” (n − 1) »2-4}. This equation to the evolute is unaltered in form, if the given equation represent a curve of the nth order. We have to eliminate between its equivalent Boothian equation and two others Substitution of the value of g from (3) in (1) and (2) leads to two quadratic equations 12a ̧ (a ̧§2 + b ̧ŋ2)” ƒ2 — 4a‚b‚n2 (b ̧n2+9a ̧§3) ƒ + 9a‚2b ̧§3n2 = 0 1 = Lƒ2 + Mƒ+ N, ƒ2 {48 (b ̧ — a ̧) § (a ̧§2 + b ̧ñ3) — 4a,b ̧ (b ̧n2 +9a ̧§3)} 2 1 +ƒ{72a ̧a ̧b ̧§n2 – 12a‚b‚§ (b ̧ŋ2+3a ̧§2) + 18a,*b ̧§1} Their eliminant determines the evolute (Ln — Nl)2 – (Lm — Ml) (Mn — Nm) = 0. After rejecting the factor (3a, nb)", this may be reduced to the normal form §3 {12a, (a ̧n2 +b ̧§1)2 — 4a,b, ̧1§3 — 36a‚a‚b‚§n2 + 9a ̧2b ̧n"} 3 3 + 4 (§2 + n2)2 {a ̧b ̧ (§a − 2n3) + 3a ̧1§n* + 3a ̧b ̧§ (2n2 — §3)} = 0. 13. To determine the evolute of a cubic curve of the third order with double points, referred to trilinear coordinates. If the equation to the curve be selected as 3.3 a1a3a3 + 3a ̧àaca3y + 3b2b3cß3y = 0, the tangential equation to its evolute may be deduced from that given in § 10 by interchanging p, q, r with P, Q, R. This conclusion is inferred from the result of the calculation in the preceding paragraph, wherein a superfluous factor is rejected, and from the general consideration of the degree of the equation of an evolute and of its spherical interpretation. 14. To find the equation to the evolute of a cubic of the third class referred to three-point tangential coordinates. This equation can be expressed in a symmetrical form, after the process adopted in §2, and afterwards be reduced to the normal form, in which the foci of the cubic and its |