XXXVI 



tributions can be indicated : it must be understood that, here as 

 elsewhere, the statement does not pretend to be a complete account. 



It is an old-established property that two curves of degrees m and n 

 cnt in mn points, but that it is not possible to draw a curve of degree n 

 ihrough any mn arbitrarily selected points on a curve of degree m. 

 As early as 1843, Cayley extended the property and showed that when 

 a curve of degree r higher than either m or n is to be drawn through 

 the mn points common to the two curves, they do not count for mn 

 conditions in its determination, but only for a number of conditions 

 smaller than mn by -|(m-h n r l) (m + n r 2). A single addition 

 was made to the theorem by Bacharach* in 1886 taking account of 

 the case when the undetermining points lie on a curve of degree 

 m+nr 3; with this exception the algebraical problem was com- 

 pletely solved by Cayley in his original paper.f The result is often 

 called Cayley's intersection-theorem. 



Another geometrical research of fundamental importance was em- 

 bodied by him in a memoirj ' On the higher singularities of a 

 plane curve,' published in 1866 : it is there proved that any singu- 

 larity whatever on a plane algebraical curve can be reckoned as 

 equivalent to a definite number of the simple singularities constituted 

 by the node, the ordinary cusp, the double tangent and the ordinary 

 inflexional tangent. The theory has, since that date, been developed 

 on lines different from Cayley's owing to its importance in other 

 theories, such as Abelian functions, variety in its development has 

 proved both necessary and useful ; but it was Cayley's investigations 

 in continuation of Plucker's theory that have cleared the path for the 

 later work of others. 



The classification of cubic curves had been effected by Newton in 

 his tract ' Ennmeratio linearum tertii ordinis,' published in 1704 : 

 and six species had been added by Stirling and Cramer, the total 

 then being 78. Plucker effected a new classification in his ' System 

 der analytischen Geometric,' published in 1835 : his total number of 

 species is 219, the division into species being more detailed than 

 Newton's. Cayley re-examined the subject in his memoir ' On the 

 classification of cubic curves,' expounding the principles of the two 

 classifications and bringing them into comparison with one another ; 

 and entering into discussion with full minuteness, he obtains the 

 exact relation of the two classifications to one another a result of 

 great value in the theory. 



To the theories of rational transformation and correspondence he 

 made considerable additions. Two figures are said to be rationally 



* 'Math. Ann.,' vol. 26 (1886), pp. 275299. 



t ' C. M. P.,' Tol. 1, No. 5 ; < Camb. Math. Journ.,' vol. 3 (1843), pp. 211213. 

 t 'C. M. P.,' vol. 5, No. 374; 'Q,uart. Math. Journ.,' TO!. 7 (1866), pp. 212 223 

 ' C. M. P.,' vol. 5, No. 350 ; ' Camb. Phil. Trans.,' vol. 11 (1864), pp. 81128. 



