MATHEMATICS 3 



transitive Abelian sub-group has a self-conjugate sub-group, 

 except possibly when the operations of the Abelian sub-group 

 are all of the same prime order. 



Vahlen extended Newton's theorem, that the first n sums of 

 powers of the roots of an equation of degree n form a funda- 

 mental system for symmetric functions of the roots, by showing 

 that if any m of the numbers 2, 3, ... n be chosen and the 

 sums Snt Sj„, SjOT . . . be set aside then the first n left form a funda- 

 mental system. B. von Ludwig {Math. Ann., 83, 1921, 67-9) 

 now shows that if g be any positive integer, « > ^ )> J (« — i ), 



and from the {g + i) sums s,+i, s^+g • • • ^io+i any «-^ Be selected, 

 then they, together with s^, Sg, . . . s„ form a fundamental 

 system. 



C. W. Gilham {Proc. Lond. Math. Soc, 20, 1921, 326-8) 

 extends to forms involving any number of variables the theorems 

 that the Jacobian of a Jacobian is reducible and that the pro- 

 duct of two Jacobians is expressible as a sum of three term 

 products. 



Lagrange's series gives only one root of the equation 

 X — a — uf(x) = o ; M. Kossler (Proc. Lond. Math. Soc, 20, 1921, 

 365-73) obtains series for the other roots. 



An equation of the wth degree with rational coefficients is 

 said to be without " affect " if its Galois group with respect to 

 the field of rational numbers is equivalent to the symmetrical 

 group of the nth degree. Hilbert proved in 1892 that such 

 equations must exist for any n ; I. Schur (Jahresber. d. deut. 

 Math. Verein, 29, 1920, 145-50) gives examples. 



L. Berzolari {Rend. Lombardo, 54, 1921, 225-38) has a 

 note on the involution determined by a binary form of the 

 fourth order and its Hessian. 



By means of orthogonal matrices, J. Radon {Abhl. Math. 

 Seminar Hamburg, 1, 192 1, 1-14) determines the maximum value 

 of p for a given n, such that the equation {x-^ + x^ + . . . + x^) 

 {yi + y^ + ■ ' ■ + yn) = z^ + z^ + . . . z^ can be satisfied by 

 bilinear functions z oi x and y. li n = 2*"+^ n^ (/^ = o, i, 2, 3 ; 

 n' odd) then /> = 2^ + 8a, in agreement with previous results of 

 Hurwitz. 



There are not many branches of pure mathematics in which 

 the pioneers are for the most part of the Anglo-Saxon race, but 

 one such branch is the algebra of hypercomplex numbers. 

 Here the important names are Hamilton, Cayley, Sylvester 

 and, more recently, the Americans Dickson and Wedderburn ; 

 the best introduction to the subject, with full bibliographical 

 information, is the Cambridge Tract by Dickson : Linear 

 Algebras. The reason for the comparative neglect of this sub- 

 ject on the Continent is perhaps a general conviction that it 

 has no applications. O. Scorza has shown, however, that it is 



