6 SCIENCE PROGRESS 



metric determinants. Sir Thomas Muir (Proc. Roy. Soc. Edin- 

 burgh, 191 8, 38, 219-25) draws attention to a notation that 

 promises to be helpful in investigating the quadratic relations 

 between the determinants of a 4-by-8 array. 



G. A. Miller (Trans. Amer. Math. Soc. 191 8, 19, 299-304) 

 extends, especially along the line of substitution groups, his 

 theorems of 191 5 (ibid. 16, 399) relating to sets of independent 

 generators of a group of finite order. Miller (Amer. Journ. 

 Math. 1 91 9, 41, 1-4) determines fundamental properties of the 

 groups generated by two operators whose relative transforms 

 are equal to each other. 



L. R. Ford (Proc. Edinburgh Math. Soc. 191 6-1 7, 35 ; Re- 

 search Paper, 191 7, No. 6) explains the fact that more small 

 money is required for the transaction of business in consequence 

 of a uniform rise of prices. 



J. L. Walsh (Trans. Amer. Math. Soc. 191 8, 19, 291-8) 

 starts from Bocher's theorem (1904) concerning a problem in 

 statics and its relation to certain algebraic invariants, obtains 

 theorems on the location of the roots of the Jacobian of two 

 binary forms, and applies his results to the roots of the deriva- 

 tive of a rational function. 



Analysis. — K. Ananda Rau (Proc. Lond. Math. Soc. 1919, 

 17, 334-6) completes the proof of a theorem stated, but not 

 fully proved, by G. H. Hardy (ibid. 12, 174-80) as to the con- 

 clusion of the convergence of a certain series from its summa- 

 bility. 



There is a very thorough review of S. Pincherle's Lezioni 

 di calcolo infmitesimale of 191 5 by E. Bortolotti (Boll, di bibl. 

 e st. delle sci. mat. 191 8, [2] 1, 46-51). 



W. H. Young (Proc. Lond. Math. Soc. 191 8, 17, 195-236) 

 obtains many important results on the convergence of the 

 derived series of Fourier series by arguments based on reasoning 

 analogous to that which he has already employed in a previous 

 communication to the Society mentioned. This paper is con- 

 nected with two of 191 7 (Proc. Roy. Soc. A, 93, 276-92, 455-67), 

 with Young's (ibid. 191 8, A, 95, 22-9) paper on the Cesaro con- 

 vergence of restricted Fourier series, and with Young's (Proc. 

 Lond. Math. Soc. 1919, 17, vi-ix, 353-66) very important proof 

 that there is a certain sub-class (" R. F. series ") of restricted 

 Fourier series such that almost all the more important pro- 

 perties of Fourier series hold good for R. F. series in the in- 



