( SIB ) 



sentatiüu as the one used here, a horizontal line connecting A and 

 B-as was the opinion of Kipping and Pope, but may quite as well 

 show a maximum or a minimum, which then however lies at 50 

 pCt. I even suspect such to be the case in camphersulfonic chlorid 

 and carvontribromide. 



Accordirg to these views, neither a higher nor a lower melting- 

 point furnishes a proof on the nature of an inactive substance, but 

 the stud}' of the entire melting-line does. 



A single curve serves in case of mixed crystals, two curves in 

 case of an inactive conglomerate, three in case of a combination. 



Other remarkable phenomena may still present themselves, in 

 case transformations of the combination, mixture or conglomerate 

 appear after the congelation. 



Mathematics. — ".4 geometrical interpretation of the invariant 

 11 {ah)" of u binari/ form a^" of even degree". By Prof. 



n+l 



P. H. SCHOUTE. 



With regard to the creation of the beautiful theory of the in- 

 variants undoubtedly very much is due to Sylvester as well 

 as to Akonhold, Boole, Brioschi, Caylry, Clebsch, Gordan, 

 HerMITE and others. As early as 1851, indeed, he developed in his 

 treatise: "O/i a remarkable discovery in the theory of canonical forms 

 and of hyper determinants" {Phil. Mag., Vol. II of Series 4, p. 

 3f)l — 410) the foundation upon which the theory of the canonical 

 forms is based. The principal contents consist of the proof of two 

 theorems. According to the first the general binary form of the odd 

 degree 2n — 1 can always be written in a single way as the sum 

 of the 2n — 1«' powers of n binary linear forms; according to the 

 second the binary form of the even degree 2 n can be written as 

 the sum of the 2 «"^ powers of n binary linear forms — and in 

 that case in a single way too — only when a certain invariant 

 vanishes. For this invariant witli which we shall deal here compare 

 a. 0. Gundelfinger's treatise in the '^Journ. f. Math.", Vol. 100, 

 p. 413—424, 1883, and Salmon's "Modern higher algebra", A^^ eA., 

 p. 156, 1885. So the theory of invariants of a certain form of any 

 kind is ruled by the question about the minimum number of ho- 

 monymous powers of linear forms by which it can be represented. 

 (Compare a.o. Reye in the "Journ. f. Math.'\ Vol. 73"^ p. 114—122). 

 With this the theory of involutions of a higher dimension and order 

 are closely allied. Likewise theorems pre deduced from it relating to 



