CORRESPONDENCE (Lat. coin-, together + to answer, from IT-, back spondcre, to promise). A term used in mathematics to certain reciprocal relations. If each in dividual of one group of objects boars a certain relation to a definite number of individuals of another group, and a definite number of individu als of the first group bears the same relation to each individual of the second group, there is said to be a correspondence between the objects of the groups. if I of the first group corresponds to B of the 2d, and 1 of the second group corre sponds to A of the 1st, the relation is called an A to B correspondence. If A = B = 1, it is called a 1 to 1 ( = 1) correspondence; e.g. two numbers are said to be equal when there exists a 1 to I correspondence between their units. In geometry the simplest cases of 1 to 1 cor respondence, or 'conformal representations,' are furnished by two planes superposed one upon the other. here to every point of the first figure there corresponds one and only one point of the second figure, and to every point of the second figure there corresponds one and only one point of the first. The simplest ease of Chasles's (q.v.) corre spondence formula may be stated thus: If two ranges of points Ti., and lie upon a straight, line so that to every point x of R, there corre spond in general a points y of R,„ and again to every point y of It„ there always correspond 13 points x of It„ the configuration formed from R, and has (a + 13) coincidences, or there are (a (3) times in which a point, x corresponds with a point x. From these linear transforma
tions Poncelet, Plucker, Magnus, Steiner, passed to the quadratic where they first investigated 1 to 1 correspondence between two separate planes. The 'Steiner projection' (1832) em ployed two planes E, and together with two straight lines y, and p. not co-planar. if we draw through a point P, or of E, or the straight line a., or ,r, which cuts g, as well as g„ and determines the intersection X, or X„ with E, or E„ then are P, and X, and P, and X, corre sponding points. In this manner to every straight line of the one plane corresponds a collie section in the other. In 1847 Plficker had deter mined a point upon I he hyperboloid of one sheet, like fixing a point in the plane. by the segments cut off by two fixed generators upon the two generators passing through the point. This was an example of a uniform representation of a sur face of the second order upon the plane. Corre spondence relative to surfaces has been studied. by Chasles, Clebsch, Cremona. Cayley, and others. In space of three dimensions, only a beginning has been made in the development of this theory. Consult: Schubert, Kalkiil der Al.,-Ahlenden Geo metric (Leipzig, 1879) ; Klein, Fcryleichende Bctrachtungen. nix(' nencre geometrische Forsch vngen (Erlangen, 1S72) :Mains, Der baryeen trische Cahill (Leipzig, 1827).