2 ;8 SCIENCE PROGRESS 



table of logarithms of numbers recently compiled, by the use 

 of which involution and evolution are reduced to addition and 

 subtraction. These tables are likely to be very useful, for 

 fractional indices, both positive and negative, are continually 

 occurring in most branches of modern experimental science. 

 Perhaps the most striking thing in D. N. Lehmer's List of 

 Prime Numbers from i to 10, 006, 721 (Washington D.C.: Car- 

 negie Institution of Washington, 19 14, 5 dollars) is the proof 

 it gives of the accuracy of Riemann's famous formula for the 

 number of primes which are less than a given number. The 

 error is zero up to 9,050,000, and for 10,000,000 it is only -f 87. 

 The errors fluctuate in sign, and on this account alone Rie- 

 mann's formula is superior to those of Legendre and Chebichev. 

 In the theory of numbers, H. C. Pocklington (Proc. Camb. 

 Phil. Soc. 191 5, 18, 29), is occupied with a method for deter- 

 mining whether a large number is prime or composite. 



Algebra. — Major P. A. MacMahon, in two papers (Trans. 

 Camb. Phil. Soc. 22, 55, 101) of 1914 and 191 5, (1) defined and 

 investigated permutation-indices of a new kind ; (2) exhibited 

 in their categories the invariants, algebraic and operational, of 

 the Halphenian homographic substitution and the transforma- 

 tion of linear differential operators in general. 



Prof. W. Burnside (Proc. Lond. Math. Soc. 191 5, 14, 251) 

 completely solves what Cayley has called the problem of 

 cyclotomic quinquisection — the determination of the system 

 of relations expressing any rational function of the roots of a 

 certain Abelian equation with a cyclical group as a linear 

 function of the roots. To the theory of groups also belong 

 W. E. H. Berwick's researches on the condition that a quintic 

 equation should be soluble by radicals (ibid. 301). 



Analysis and Theory of Functions. — Tetsuzo Kojima (Science 

 Reports of Tohoku Imperial University, 3) gives a method for 

 obtaining a better approximation to the graphical solution of 

 the ordinary linear differential equation of the first order 

 between two variables than that given by Czuber, and further 

 applies graphical methods of a similar character to the approxi- 

 mate solution of certain other first-order equations, including 

 the general equation of the first order and second degree. 



Some years ago, Prof. H. F. Baker gave a general method 

 for the solution of linear differential equations ; and, in a 

 paper read to the Royal Society on June 17, 191 5, he showed 





