RECENT ADVANCES IN SCIENCE 95 



be found which will coincide with a given function f(x) at any 

 n + i interior points of the interval. 



Prof. W. F. Osgood (Trans. Amer. Math. Soc. 1916, 17, 1) 

 decides four open questions in the theory of analytic functions 

 of several complex variables by means of simple examples. 



Goursat pointed out in 1891 that Lagrange's classification 

 of the integrals of a partial differential equation of the first 

 order into three groups is incomplete, and since then Forsyth 

 has repeatedly drawn attention to the fact. L. L. Steimley 

 (Amer. Journ. Math. 191 5, 37, 359) gives a new and complete 

 classification of all the integrals of the linear non-homogeneous 

 equation, and a means is developed whereby all the elusive 

 " special " integrals can be readily determined as soon as the 

 Lagrange general integral is known. 



With respect to difference equations, Tomlinson Fort (ibid. 

 43) shows how a method developed by Liapounoff in 1902 

 for the linear differential equation of the second order can be 

 extended to the difference equation in which the independent 

 variable is restricted to integral values; C. E. Love (ibid. 

 1 916, 38, 57) adapts the methods of Dini (1899, 1900) for the 

 integration of linear differential equations for large values of 

 the independent variable and the parallel investigation of Ford 

 (1907) for linear difference equations, to the study of somewhat 

 more general classes of equations; and R. D. Carmichael 

 (ibid. 185) gives a new method for investigating the solutions of 

 linear homogeneous difference equations. 



According to R. B. Robbins (ibid. 1915, 37, 367) Lagrange's 

 analytical formulation of the calculus of variations probably 

 prevented the early recognition of a close connection between 

 its problem and the ordinary problem of maxima and minima. 

 In Robbins 's paper the algebraic problem of minimising a sum 

 is compared with the transcendental problem of minimising a 

 definite integral, with interesting results. 



Cauchy and others treated familiar functions by their func- 

 tional equations, and E. B. Van Vleck and F. H'Doubler 

 (Trans Amer. Math. Soc. 1916, 17, 9) study analogously certain 

 more complicated functional equations for the Theta functions. 



When we consider, for example, the bending of an elastic 

 bar under first increasing and then decreasing weights, we 

 arrive at the conclusion that the future states of a body do 

 not depend merely on the present state and the immediately 



