366 SCIENCE PROGRESS 



sion of the product of two oblong arrays, in which the form 

 of the expansion given is an aggregate of single determinants. 

 The relation with the form given by Binet and Cauchy in 1812 

 as a sum of products of pairs of determinants is explained, and 

 an historical remark added. 



W. A. Manning (Amer. Journ. Math. 191 7, 39, 281-310) 

 discusses primitive groups of class 15. 



E. Kircher (ibid. 272-80) studies some of the properties of 

 a finite algebra whose elements combine by addition and 

 multiplication, subject to the commutative, associative and 

 distributive laws. The paper is closely related to investiga- 

 tions by H. S. Vandiver (1912), L. E. Dickson (1905) and 

 A. Fraenkel (191 5). 



Analysis. — P. J. Daniell {Amer. Math. Monthly, 191 7, 24, 

 109-13) gives three rules of quadrature developed from Euler's 

 summation formula, of which two seem to be new. 



E. V. Huntington (ibid. 271-5) starts from the ordinary 

 " intuitive " process used by applied mathematicians in " setting 

 up " an integral — for example, in the problem of finding the 

 attraction of a rod — and finds that a sufficient condition that 

 the process referred to leads to a correct result is that the 

 functions in the integrand are continuous ; it is not necessary 

 to consider any questions of uniformity. The condition seems 

 very important from the point of view of principles. Thus the 

 theorem known as " Duhamel's " can be dispensed with — a 

 fortunate circumstance, as the theorem is sometimes false. The 

 paper is connected with papers by W. F. Osgood (1903), R. L. 

 Moore (1912), G. A. Bliss (1914), and an interesting suggestion 

 by a student (B. Graham, ibid. 265-71). 



E. B. Van Vleck (Trans. Amer. Math. Soc. 191 7, 18, 326-30) 

 shows that a general proof of the " momental " theorem in 

 integration of Haskins and Dunham Jackson (see Science 

 Progress, 191 6, 11, 268) can be obtained by reduction to 

 Stieltjes's theorem of moments. Incidentally it appears that 

 every Lebesgue integral can be thrown into the form of a 

 Stieltjes integral. 



W. H. Young (Proc. Lond. Math. Soc. 191 7, 16, 175-218) 

 defines a new kind of integral which is non-absolutely conver- 

 gent and not necessarily continuous. Absolutely convergent 

 integrals, taken with respect to discontinuous functions of 

 bounded variation, are in general discontinuous functions ; 



