XXXV11 



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 transfor- 

 mations 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 investi- 

 jratioiis on the bitangents of plane curves and, in particular, on the 

 '28 bitangents of a non-singular quartic ; his developments of Plucker'ft 

 conception of foci ; his discussion of the osculating conies of curves, 

 and of tho sextactic points on a plane curve (these are the places 

 where a conic can be drawn through six consecutive points) ; his con- 

 tributions 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 Philo- 

 sophical Society, a curve-tracing mechanism connected with three-bar 

 motion. 



All the preceding results belong to plane geometry ; no less im- 

 portant 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 memoirf ' On the Triple Tangent Planes 

 of Surfaces of the Third Order,' published in 1849, after a correspon- 

 dence 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 



* In this connection a report by Brill and Noether, " Bericht uber die 

 Kntwicklung der Theorie der algebraischen Functionen in alterer und neuerer 

 Zeit" ('Jahresber. d. Deutschen Mathem.-Vereinigung,' vol. 3, 1894) will be 

 found particularly the sixth and the tenth sections to give a very valuable 

 resume of the theory and its history. 



f ' C. M. P.,' vol. 1, No. 76; 'Camb. and Dubl. Math. Journal,' vol. 4 (18-W)), 

 pp. 118132. See also Salmon's 'Solid Geometry' (third edition, 1874), p. 46 1, 

 note. 



