RECENT ADVANCES IN SCIENCE 349 



L. H. Rice (Amer. Journ. Math. 191 8, 40, 242-62) gives an 

 extended definition of a determinant which applies to deter- 

 minants of more than three dimensions, and enables us to 

 remove the restriction of Cayley's law of multiplication and to 

 set up a new case in Scott's law of multiplication ; and gives 

 an application to transvectants. 



O. E. Glenn (Trans. Amer. Math. Soc. 191 8, 19, 109-18) 

 shows that the system of covariants of a binary cubic, when 

 transformed by a group of all linear substitutions on x lt x 2 

 whose coefficients are least positive residues with the modulus 2, 

 is finite and that the fundamental set consists of twenty 

 quantics. 



A. Ostrowski (Math. Ann. 191 7, 78, 94-119) finds the neces- 

 sary and sufficient conditions for the existence of a finite basis 

 with certain systems of whole functions. 



C. H. Rawlins (Amer. Journ. Math. 191 8, 40, 155-73) derives 

 complete systems of concomitants of the three-point and the 

 four-point in elementary geometry, with some applications. 



W. A. Manning (Trans. Amer. Math. Soc. 191 8, 19, 127-42) 

 gives the third part of his researches on the order of primitive 

 groups : the former parts were published in 1909 and 191 5. 



P. Fatou (Compt. rend. 191 7, 164, 806-8) has a note on 

 rational substitutions. 



H. B. A. Bockwinkel (Proc. K. Akad. Amst. 19, 1 100-14) 

 gives an English translation of his paper on transmutations in 

 the Verslagen (cf. Science Progress, 191 8, 13, 10). 



Analysis. — K. P. Williams (Amer. Math. Monthly, 191 8, 25, 

 246-9) and E. W. Chittenden (ibid. 249-50) find various pro- 

 perties of functions which approach a limit at every point of 

 an interval. 



H. Blumberg (Bull. Amer. Math. Soc. 191 8, 24, 381-3) 

 proves a general theorem on semi-continuous functions, which 

 is analogous to those of G. C. Young (1 914) and A. Denjoy (191 5) 

 on sets of points where the four derivatives of a given continu- 

 ous function are identical. 



A. Loewy (Math. Ann. 191 7, 78, 1— 51 ) writes on matrices 

 and differential complexes. 



A. Denjoy (Ann. de I'ec. norm. [3], 33, 127-222) gives the 

 first part of a continuation of some work of his published in 

 191 5. Here he solves the problem of determining the primitive 

 of a function which is known to be a derivative. 



