754 



SCIENCE 



[N. S. Vol. XXXIII. No. 855. 



erality as much as in simplicity and one 

 could make out a whole series of possible 

 generalizations. A very simple lemma in- 

 spired by Kronecker's ideas had made this 

 result possible. 



Consider an indefinite series of forms F 

 depending upon n variables; we can find 

 among these a finite number of forms 

 F^, ■••, Fp, such that any form F of the 

 series can be equated to 

 (1) Jir = A,F^+...ApFp, 



the A's being forms depending upon the 

 same variables. This is a consequence of 

 the fundamental notion of the modulus 

 introduced by Kronecker. This means, in 

 Kronecker's language, that the divisors 

 common to many moduli, even were they 

 infinite in number, are submultiples of one 

 of them which is their greatest common 

 divisor, and in geometric language (sup- 

 posing four variables and regarding them 

 as homogeneous coordinates of a point in 

 space) that the aggregate of points com- 

 mon to an infinite number of algebraic 

 surfaces is composed of a finite number of 

 isolated points and a finite number of skew 

 algebraic curves. 



But this is not all; suppose the F's are 

 the invariants of a system of forms and 

 the A's functions of the eoeiificients of 

 these forms. 



We may always suppose that the A 's are 

 also invariants, otherwise we could per- 

 form an arbitrary linear transformation 

 upon the forms. Then in the relation (1) 

 thus transformed would appear the coeffi- 

 cients of this transformation. In applying 

 to the relation (1) transformed a certain 

 train of successive differentiations (the 

 differentiations are performed with respect 

 to the coefficients of the linear transforma- 

 tion) we reach a relation of the same form 

 as (1) but where the A's are invariants. 

 From this the proof of Gordan's theorem 

 follows immediately. 



But this is not all; among these funda- 

 mental invariants there is a certain num- 

 ber of relations called syzygies. All the 

 syzygies can be deduced from a finite num- 

 ber of them by addition and multiplica- 

 tion. Among these fundamental syzygies 

 of the first order there are syzygies of the 

 second order, which can also be obtained 

 from a finite number of them by addition 

 and multiplication, and so on. 



Hilbert gets this result from a general 

 theorem of algebra. Consider a system of 

 linear equations of the form 



where the F's are given forms and the X's 

 unknown forms homogeneous in regard to 

 certain variables; the study of the solu- 

 tions of this system and of the relations 

 which connect them leads to the considera- 

 tion of a series of derived systems con- 

 tinued until we reach a derived system 

 which no longer admits of any solution. 

 Thus it was that Hilbert was led to deter- 

 mine and to study the number X{B) of 

 distinct conditions which a form of degree 

 R should satisfy to be congruent to zero 

 with regard to a given modulus. 



But to complete the theory it was not 

 enough to establish the existence of a sys- 

 tem of fundamental invariants; it was 

 necessary to give the means of actually 

 forming this, and this problem was made 

 by our author to depend upon a question 

 which connects it with the theory of whole 

 algebraic numbers extended to integral 

 polynomials. 



The problem is thus broken up into three 

 others. 



1. To find invariants Jm as functions of 

 which all the others can be expressed in 

 algebraic and integral form, that is to say, 

 such that any invariant / satisfies an alge- 

 braic equation 



Jk + G^Jk-l + G,J!<r2 + . . . -\-G^^J + G,c = 0, 



