Mat 19, 1911] 



SCIENCE 



755 



the G's being polynomials integral with 

 regard to Jm- 



2. To find invariants as functions of 

 which all the others can be expressed ra- 

 tionally. 



3. To find invariants as functions of 

 which all the others can be expressed in 

 rational and integral form. 



Of these three problems the first is the 

 most difficult. If it be supposed solved, 

 the aggregate of invariants presents itself 

 as an algebraic corpus, and the first step to 

 make is to determine the degree of this 

 corpus; this it is at which Hilbert arrives 

 at least for binary forms by evaluating in 

 two different ways the number <^(o-) of 

 invariants linearly independent of degree 

 or, or rather the asymptotic value of this 

 numeric function ^(o-) for o- very great. 



The first problem once solved, the solu- 

 tion of the other two goes back to a classic 

 question of the arithmetic of polynomials 

 and of the theory of algebraic corpora. 

 The question is then to find the funda- 

 mental invariants by whose aid all the 

 others can be expressed in algebraic and 

 integral form. 



With this purpose Hilbert remarks that 

 these are those which can not be annulled 

 without annulling all the others. So we 

 see that the search for these fundamental 

 invariants will be singularly facilitated by 

 the study of null forms, that is to say, of 

 those whose numeric coefficients are chosen 

 in such a way that the numeric values of 

 all the invariants may be null. 



In the case of binary forms, the null 

 forms are those which are divisible by a 

 sufficiently high power of a linear factor; 

 but in the other cases the problem is more 

 delicate. Our author first establishes a 

 certain number of theorems. 



Consider a form with numeric coeffi- 

 cients and its transform by any linear sub- 

 stitution; the coefficients of this transform 



will be integral polynomials with regard to 

 the coefficients of the substitution. If the 

 determinant of the substitution is an alge- 

 braic and integral function of these in- 

 tegral polynomials, the proposed form is 

 not a null form. In the contrary case, it 

 is a null form. 



Consider, on the other hand, the trans- 

 forms of a form by a linear substitution 

 depending upon an arbitrary parameter t 

 and in such a way that the coefficients of 

 this substitution are series developable in 

 positive or negative integral powers but 

 increasing with this parameter. If it be a 

 question of a null form, we can choose a 

 substitution of this kind of such a sort that 

 its determinant becomes infinite for i = 0, 

 while the coefficients of the form trans- 

 formed remain finite. Hilbert shows that 

 this condition is necessary in order that 

 the proposed form may be null, and it is 

 evident, moreover, that it is sufficient. To 

 each null form corresponds therefore one 

 and perhaps several linear substitutions 

 having the enunciated property. This set- 

 tled, our author proves that, starting from 

 any null form, we can by a linear trans- 

 formation, transform it into a canonic null 

 form. A form is called canonic when the 

 linear substitution which corresponds to it 

 and which possesses in relation to it the 

 property we have just stated is of the 

 simple form 



The investigation of null forms is thus 

 made to depend on that of canonic null 

 forms which is much more simple. We 

 find that the canonic null forms are those 

 in which certain terms are lacking; and 

 the determination of the terms which 

 should be lacking can easily be made, 

 thanks to a simple geometric scheme. 



