5i8 SCIENCE PROGRESS 



the quadrin variant. This result is given in a recent paper 

 by A. Ostrowski and I. Schur {Math. Zs., 31, 1922, 81-105). 



Sir Thomas Muir (Proc. R.S. Edin., 42, 1922, 242-7) simpU- 

 fies and so brings out the real significance of a theorem of 

 Frobenius' on an aggregate of products of minors of a rectan- 

 gular array. 



H. Stenzel {Math. Zs., 15, 1922, 1-25) investigates the 

 problem of finding the most general matrix X which satisfies 

 the equation CX = XC', where C is a given square matrix of 

 n rows. This is equivalent to finding the most general sym- 

 metrical and skew matrices satisfying the equation. The 

 author gives them explicitly and obtains conditions for C in 

 order that their determinants shall not be zero ; he also finds 

 necessary and sufficient conditions for the representation of 

 a matrix as the product of two symmetrical, of two skew, or 

 of a symmetrical and a skew matrix. 



R. Mehmke {Crelle, 152, 1922, 33-9) generalises the work of 

 K. Bohm on the product of matrices. 



W. H. Young {Proc. L.M.S., 21, 1922, 75-94, 161-90) 

 continues his investigations on the transformation theory of 

 double integrals, showing in particular that practically the 

 formula for an area is true in all the cases in which the double 

 integral of the Jacobian is known to exist. 



The first rigorous proof of Cauchy's Theorem without the 

 use of a double integral was given by Malmsten in 1 865 ; Mittag- 

 Leffler later (1873) gave a simpler and more direct proof which 

 also included Laurent's Theorem and required fewer assump- 

 tions for the function than did Cauchy's, He now {Crelle, 152, 

 1922, 1-5 ; Arkiv for Mat., 17, 1922, No. 6) returns to the 

 subject, analysing the conditions of the various proofs and 

 pointing out that the assumptions of Goursat's proof of 1884 

 are not identical with, do not include nor are included by his 

 own. 



A. Kienast {Crelle, 152, 1922, 109-19) considers the repre- 

 sentation of analytic functions by means of definite integrals 

 in connection with their asymptotic expansions. 



It was shown by Eisenstein in 1852 that if an algebraic 

 function can be expanded as a power series with rational co- 

 efficients, these coefficients will in general be integers, or, more 

 precisely, a necessary condition for the series 



/(^) = 7 + 7^ + . . •+7^n + . • . 



to represent an algebraic function is that a positive integer 

 T can be found so that 



/(T^) -/(o) =^^ + . . . +^2" -h . . . 



