352 SCIENCE PROGRESS 



At the basis of Frechet's " functional calculus " (1906) is a 

 generalisation of the distance between two points, which 

 associates with each pair of elements of an abstract class a real 

 non-negative number. He was thus able to retain the more 

 important theorems of the theory of sets of points and of real 

 functions. A. D. Pitcher and E. W. Chittenden {Trans. Amer. 

 Math. Soc. 191 8, 19, 66-78) simplify the foundations of Frechet's 

 theory in some interesting respects. 



M. Frechet (ibid. 53-65), starting from the fact that we 

 can, in many .cases, define convergent sequences and their 

 limits in a class of abstract elements, determines the condition 

 to which a choice must be subject in order that we may define 

 on this class a distance such that convergence already defined 

 will not be altered when it is defined by means of this distance. 



R. Schauffler (Math. Ann. 1917, 78, 52-62) treats limit- 

 questions and other questions in the theory of iterated functions, 

 and F. Tricomi {Giorn. di Mat. 1916, 54, 35-42) discusses 

 iteration of functions of a line. 



W. C. Graustein (Bull. Amer. Math. Soc. 191 8, 24, 473-7) 

 gives a theorem on isogenous complex functions of curves. 



T. Fort (ibid. 330-4) gives some theorems of comparison and 

 oscillation. 



O. D. Kellogg (Amer. Journ. Math. 191 8, 40, 145-54, 225-34) 

 connects with the theory of integral equations the condition 

 previously found by him (Science Progress, 1916, 11, 94-5) 

 for many of the oscillating properties of the more common sets 

 of orthogonal functions. 



R. G. D. Richardson (ibid. 283-316) makes some contribu- 

 tions to the study of oscillation properties of the solutions of 

 linear differential equations of the second order. 



J. Horn (Archiv der Math. 191 6, 25, 137-48) shows that the 

 theory of non-linear difference equations has a certain analogy 

 with that of non-linear differential equations. 



Various points in the theory of the dynamical equations 

 are dealt with by F. Engel (Gott. Nachr. 191 6, 270-5), E. Vessiot 

 (Compt. rend. 191 7, 165, 99-102), and K. Bohlin (Journ. de 

 Math. 191 6, [7], 2, 173-200). 



C. E. Wilder (Trans. Amer. Math. Soc. 191 8, 19, 157-66) 

 discusses some problems in the theory of a system consisting of 

 an ordinary linear differential equation and auxiliary conditions 

 involving linearly the values of the solution and its derivatives 



