BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
XXXIX 
To the theories of rational transformation and correspondence he made considerable 
additions. Two figures are said to be rationally transformable into one another when 
to a variable point of one of them corresponds reciprocally one (and only one) variable 
point of the other. The figure may be a space or it may be a locus in a space. 
Rational transformations between two spaces give rational transformations between loci 
in those spaces; but it is not in general true that rational transformations between 
two loci necessarily give rational transformations between the spaces in which those loci 
exist. There is thus a distinction between the theory of transformation of spaces and 
the theory of correspondence of loci. Both theories have occupied many investigators, 
the latter in particular; and Cayley’s work may fairly be claimed to have added much 
to the knowledge of the theory as due* to Riemann, Cremona and others. 
Further, there may be singled out for special mention, his investigations on the 
bitangents of plane curves, and, in particular, on the 28 bitangents of a non-singular 
quartic; his developments of Pliicker’s conception of foci; his discussion of the osculating 
conics of curves, and of the sextactic points on a plane curve (these are the places 
where a conic can be drawn through six consecutive points); his contributions to the 
geometrical theory of the invariants and covariants of plane curves; and his memoirs 
on systems of curves subjected to specified conditions. Moreover, he was fond of making 
models and of constructing apparatus intended for the mechanical description of curves. 
The latter finds record in various of his papers; even so lately as 1893 he exhibited, 
at a meeting of the Cambridge Philosophical Society, a curve-tracing mechanism con 
nected with three-bar motion. 
All the preceding results belong to plane geometry; no less important or less 
numerous were the results he contributed to solid geometry. The twenty-seven lines 
that lie upon a cubic surface were first announced in his memoirj*, “On the triple 
tangent planes of surfaces of the third order,” published in 1849, after a corre 
spondence between Salmon and himself. Cayley devised a new method for the analytical 
expression of curves in space by introducing into the representation the cone passing 
through the curve and having its vertex at an arbitrary point. Again, by using 
Pliicker’s equations that connect the ordinary (simple) singularities of plane curves, he 
deduced equations connecting the ordinary (simple) singularities of the developable surface 
that is generated by the osculating plane of a given tortuous curve, and, therefore, also 
of any developable surface. He greatly extended Salmon’s theory of reciprocal surfaces; 
and resuming a subject already discussed by Schläfli he produced! in 1869 his “Memoir 
on cubic surfaces,” in which he dealt with their complete classification. Many of his 
memoirs are devoted to the theory of skew ruled surfaces, or scrolls as he called them. 
Our knowledge of geodesics, of orthogonal systems of surfaces, of the centro-surface of 
an ellipsoid, of the wave-surface, of the 16-nodal quartic surface, not to mention more, 
* In this connexion a report by Brill and Noether, “Bericht über die Entwicklung der Theorie der 
algebraischen Functionen in älterer und neuerer Zeit” (Jahresber. d. Deutschen Mathem.-Vereinigung, vol. in. 
1894) will be found—particularly the sixth and the tenth sections—to give a very valuable resume of the 
theory and its history. 
t C. M. P. vol. i. No. 76; Camb. and Dull. Math. Jour. vol. iv. (1849), pp. 118—132. See also Salmon’s 
Solid Geometry (third edition, 1874), p. 464, note. 
t C. M. P. vol. vi. p. 412; Phil. Trans. (1869), pp. 231—326. 
C. VIII. 
/