60 



SCIENGE. 



[N. S. Vol. IX. No. 211. 



the source of his mistake, the unsuspected and 

 so neglected interdependence of certain syzygies, 

 and devoted his Ninth Memoir on Quantics (7th 

 April, 1870) to the correction of his error and 

 a further development of the theory iu the light 

 of Gordan's results. 



The whole of this primal theory of invariants 

 may now be regarded as a natural and elegant 

 application of Lie's theory of continuous groups. 

 The differential parameters, which in the or- 

 dinary theory of binary forms enable us to cal- 

 culate new invariants from known ones, appear 

 in a simple way as differential invariants of 

 certain linear groups. The Lie theory may be 

 illustrated by a simple example. 



Consider the binary quadratic form 



/ = ao.x-2 + 2ai.T?/ -\-a.,y^-. 

 Applying to/ the linear transformation 



( 1 ) x = axi + Py', y = yx' + Sy', 



we obtain the quadratic form 



/' = a\x'' -\- 2a\x'y' + a'^y'"', 

 where the coefllcients are readily found to be 



(2) a\=a(ia, + {a6 + py)a,+yia,, 



We may easily verify the following identity : 

 a\a\ — o'^i ^ [ad — /'3}')'Hao''z — «'i)- 



Hence a^a^ — a^j is an invariant of the form 

 f. In the group theory it is an invariant of the 

 group of linear homogeneous transformations 

 (2) on the three parameters «„, «„ a.,. 



The only covariant of/ is known to be / itself. 

 Iu the Lie theory it appears as the invariant of a 

 linear homogeneovis group on five variables, x, 

 y, a^, a,, a,, the transformations being defined 

 by the equations (2), together with (1) when in- 

 verted. 



In general, the invariants of a binary form of 

 degree n are defined by a linear homogeneous 

 group on its n -r 1 coefficients, its covariants by 

 a group on n -\- S variables. 



As iu all problems in continuous groups, the 

 detailed developments are greatly simplified by 

 employing the infinitesimal transformations of 

 the groups concerned. 



It is readily proven by the group theory that 

 all invariants and covariants are expressible in 

 terms of a finite number of them. 



This result is, however, not equivalent to the 

 algebraic result that all rational integral in- 

 variants (including covariants) are expressible 

 rationally and integrally iu terms of a finite 

 number of such invariants. 



Twenty years ago, in my ' Bibliography of 

 Hyper Space and Non-Euclidean Geometry' 

 (American Journal of Mathematics, Vol. I., Nos. 

 2 and 3, 1878), I cited seven of Cayley's 

 papers written before 1873 : 



I. Chapters in the Analytical Geometry of 

 (}i) Dimensions. Camb. Math. Jour., Vol. IV., 

 1845, pp. 119-127. 



II. Sixth Memoir on Quantics. Phil. Trans., 

 Vol. 149, pp. 61-90 (1859). 



III. Note on Lobatchevsky's Imaginary 

 Geometry. Phil. Mag. XXIX., pp. 231-233 

 (1865). 



IV. On the rational transformation between 

 two spaces. Lond. Math. Soc. Proc. III., pp. 

 127-180 (1869-71). 



V. A Memoir on Abstract Geometry. Phil. 

 Trans. CLX., pp. 51-63 (1870). 



VI. On the superlines of a quadric surface in 

 five dimensional space. Quar. Jour., Vol. XII., 

 pp. 176-180 (1871-72). 



VII. On the Non-Euclidean Geometry. 

 Clebsch Math. Ann. V., pp. 630-634 (1872). 



Four of these pertain to Hyper-Space, and in 

 that Bibliography I quoted Cayley as to its 

 geometry as follows : 



"The science presents itself in two ways — - 

 as a legitimate extension of the ordinary two,- 

 and i^ree-dimensional geometries, and as a need 

 in these geometries and in analysis gener- 

 allj'. In fact, whenever we are concerned with 

 quantities connected together iu any manner, 

 and which are or are considered as variable or 

 determinable, then the nature of the relation 

 between the quantities is frequently rendered 

 more intelligible by regarding them (if only two 

 or three in number) as the coordinates of a 

 point in a plane or in space : for more than 

 three quantities there is, from the greater com- 

 plexity of the case, the greater need of such a 

 representation ; but this can only be obtained 

 by means of the notion of a space of the proper 

 dimensionality ; and to use such a representa- 

 tion we require the geometry of such space. 



An important instance in plane geometry has 



