RECENT ADVANCES IN SCIENCE 547 



This paper is related to a paper that Hitchcock published in 

 1915 {ibid. 35, 170-80; Science Progress, 1918, 12, 369). 



C. E. Wilder (Trans. Amer. Math. Soc. 191 7, 18, 415-42) 

 carries out the suggestion of Bocher (191 3) that Birkhoff's 

 (1908) results on the boundary value and expansion problems 

 for the ordinary linear differential equation of the nth order 

 with boundary conditions at two points should be generalised 

 to the equation with auxiliary conditions at more than two 

 points, and gives the proof of the convergence of the expansion, 

 which may be studied quite independently of the other results. 

 The formal development of the boundary problem and a more 

 detailed discussion of the form of series will be presented by 

 Wilder in other papers. 



J. Chazy (Acta Math. 191 6, 41, 29-69) extends to the 

 complex domain and to differential equations of any order the 

 results of Bendixson and Picard on the integrals of the equation 

 x n dy / dx=F(x, y) for small real values of x and y, where n is 

 an integer greater than 1 and F(x, y) is holomorphic and zero 

 when both x and y are zero ; and gives a further generalisation 

 and application to a case of the problem of n bodies. Then 

 Chazy gives part of his important memoir of 191 2, to which a 

 prize was awarded by the Paris Academy of Sciences, on 

 certain differential equations of the third and higher orders, 

 and shows that the integrals have transcendental singularities, 

 and are not one-valued. The results obtained in the first part 

 are here used. 



W. L. Hart (Trans. Amer. Math. Soc. 1917, 18, 125-60) 

 develops certain theorems concerning a type of real valued 

 functions of infinitely many real variables ; he then considers 

 the problem of infinite systems of corresponding ordinary 

 differential equations ; and finally discusses the fundamental 

 problem of implicit function-theory in this field. The results 

 of all three sections of the paper include as special cases the 

 corresponding theorems on functions of a finite number of 

 variables. 



G. D. Birkhoff (ibid. 199-300) gives an exposition of his 

 advances in the treatment of dynamical systems with two 

 degrees of freedom, which constitute the simplest type of non- 

 integrable problems. The researches of Hill, Poincar6, 

 Hadamard, Levi-Civita, and others have thrown great light 

 upon the subject of such systems. 



