CISSOID (from the Gr. rcunr6s,

See also:

• IVY (A.S. ifig, Ger. Epheu, perhaps connected with apium, anent)

ivy, and ethos,

See also:

• FORM (Lat. forma)

form)  , a curve invented by the Greek mathematician Diocles about 18o B.C., for the purpose of constructing two mean proportionals between two given lines; and in order to solve the problem of duplicating the cube . It was further investigated by John Wallis, Christiaan Huygens (who determined the length of any arc in 1657), and Pierre de Fermat (who evaluated the area between the curve and its asymptote in 1661) . It is constructed in the following manner . Let APB be a semicircle, BT the tangent at B, and APT a line cutting the circle in P and BT at T; take a point Q on AT so that AQ always equals PT; then the locus of Q is the cissoid . Sir Isaac Newton devised the following mechanical construction . Take a rod LMN bent at right angles at M, such that MN=AB; let the leg LM always pass through a fixed point 0 on AB produced such that OA = CA, where C is the middle point of AB, and cause N to travel along the line perpendicular to AB at C; then the midpoint of MN traces the cissoid . The curve is symmetrical about the axis of x, and consists of two infinite branches asymptotic to the line BT and forming a cusp' at the origin . The cartesian equation, when A is the origin and AB = 2a, is y2(2a—x)=x3; the polar equation is r=2a sin B tan B . The cissoid is the first positive pedal of the parabola y2+8ax=o for the vertex, and the inverse of the parabola y2=8ax, the vertex being the centre of inversion, and the semi-latus rectum the constant of inversion . The area between the curve and its asymptote is 3aa2, i.e. three times the area of the generating circle . The term cissoid has been given in modern times to curves generated in similar manner from other figures than the circle, and the form described above is distinguished as the cissoid of Diocles . A cissoid angle is the angle included between the concave sides of two intersecting curves; the convex sides include the sistroid angle .

See John Wallis, Collected

Works, vol. i . ; T . H . Eagles, Plane Curves (1885) . CIS-SUTLEJ STATES, the southern portion of the Punjab, India . The name, now obsolete, came into use in 1809, when the Sikh chiefs south of the Sutlej passed under British protection, and was generally applied to the country south of the Sutlej and north of the Delhi territory, bounded on the E. by the Himalayas, and on the W. by Sirsa district . Before 1846 the greater part of this territory as independent, the chiefs being subject merely to control from a political officer stationed at Umballa, and styled the agent of the governor-general for the Cis-Sutlej states . After the first Sikh War the full administration of the territory became vested in this officer . In 1849 occurred the annexation of the Punjab, when the Cis-Sutlej states commissionership, comprising the districts of Umballa, Ferozepore, Ludhiana, Thanesar and Simla, was incorporated with the new province . The name continued to be applied to this division until 1862, when, owing to Ferozepore having been transferred to the Lahore, and a part of Thanesar to the Delhi division, it ceased to be appropriate . Since then, the tract remaining has been known as the Umballa division . Patiala, Jind and Nabha were appointed a separate political agency in 1901 .

Excluding

Bahawalpur, for which there is no political agent, and Chamba, the other states are grouped under the commissioners of Jullunder and Delhi, and the superintendent of the Simla hill states .