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XXXV111 
BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
It is in analytical geometry, both of curves and of surfaces, that the greatest variety 
of Cayley’s contributions is to be found. There is hardly an important question in the 
whole range of either subject in the solution of which he has not had some share; 
and there are many properties our acquaintance with which is due chiefly, if not entirely, 
to him. How widely he has advanced the boundaries of knowledge in analytical geometry 
can be inferred even from the amount of his researches already incorporated in treatises 
such as those by Salmon, Clebsch and Frost; and yet they represent only a portion 
of what he has done. In these circumstances only a selection among his contributions 
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 cut in inn 
points, but that it is not possible to draw a curve of degree n through any mn arbitra 
rily 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 to 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 |(to+w — 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 + n—r — 3; with this exception the algebraical problem was com 
pletely solved by Cayley in his original paperf. The result is often called Cayley’s 
intersection-theorem. 
Another geometrical research of fundamental importance was embodied by him in 
a memoir“ On the higher singularities of a plane curve,” published in 1866: it 
is there proved that any singularity 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 Plticker’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 “ Enu- 
meratio 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 Geometrie,” 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 prin 
ciples of the two classifications and bringing them into comparison with one another; 
and entering into the discussion with full minuteness, he obtains the exact relation of 
the two classifications to one another—a result of great value in the theory. * * * § 
* Math. Ann. vol. xxvi. (1886), pp. 275—299. 
f C. M. P. vol. i. No. 5; Gamb. Math. Jour. vol. in. (1848), pp. 211—213. 
+ C. M. P. vol. v. No. 374; Quart. Math. Jour. vol. vii. (1866), pp. 212—223. 
§ C. M. P. vol. v. No. 350; Gamb. Phil. Trans, vol. xi. (1864), pp. 81—128.