RECENT ADVANCES IN SCIENCE 621 



Very many mathematicians from the time of Gauss onwards 

 have dealt with the problem of setting up criteria by means 

 of which the irreducibility of certain expressions in certain 

 domains may be seen at a glance from the character of the 

 expressions. H. Blumberg (ibid. 517-44) gives a general 

 theorem and certain immediate consequences of it which 

 include as special cases, with minor exceptions, all the results 

 hitherto obtained. A former note of 191 5 had given a summary 

 of part of this paper, but since then the author has found how 

 to unify and generalise his previous results. 



W. E. Milne (Proc. Nat. Acad. Sci., Washington, D.C., 

 191 6, 2, No. 9) gives more precise formulae than those of 

 Birkhoff when discussing asymptotic expressions in the theory 

 of linear differential equations. 



In the case of n functions of a single variable, the vanishing 

 of the Wronskian is the most familiar criterion for their linear 

 dependence. The Wronskian also plays an important part 

 in connection with the theory of a single ordinary homogeneous 

 linear differential equation of the w tb order. G. M. Green 

 (Trans. Amer. Math. Soc. 1916, 17, 483-516) generalises the 

 fundamental facts connected with these subjects to the case 

 of functions of several variables. 



E. Pascal (Giorn. di Mat. 191 5, 53, 318-48) gives a 

 systematic exposition of the analogy between the functional 

 calculus and the ordinary calculus. 



A. A. Bennett (Annals of Math. 191 5, 17, 23-60) discusses 

 the iteration of power series and of a real function of one 

 variable, and then gives some properties of certain rational 

 functions under iteration. Bennett also (ibid. 74-5) defines 

 an operation of the " third and higher grades " — addition being 

 an operation of the " first grade " and multiplication one of 

 the " second grade " — by means of the iteration of the function 



Edward B. Van Vleck (Bull. Amer. Math. Soc. 1916, 23, 

 1-1 3) gives an exceedingly interesting address on current 

 tendencies of mathematical research. His view is that " the 

 problem of the infinite set " is the most conspicuous in modern 

 pure mathematics, and his address is devoted to showing some 

 of the forms in which this problem appears. Generalisation to 

 an enumerably infinite number of variables is illustrated by the 

 passage from a set of ordinary linear algebraic equations to a 



