i 9 8 SCIENCE PROGRESS 



convergence of the series, how to compute the coefficients. It is 

 then shown that, if there exists one expansion for f(t) satisfy- 

 ing certain conditions, this expansion is unique. 



Some years ago A. Hurwitz showed how to calculate Fourier's 

 constants of the product of two functions from those of the 

 functions themselves. W. W. Kustermann (Amer. Journ. 

 Math. 191 7, 39, 113-22) solves the corresponding problem for 

 multiple Fourier's series. The work is carried through for 

 double series only, but the results and proofs admit of obvious 

 generalisation to w-tuple series. 



F. Tricomi (Giorn. di Mat. 191 6, 54, 1-9) proves a theorem 

 on the convergence of sequences formed by successive iterations 

 of a function of a real variable. 



C. A. Fischer (Bull. Amer. Math. Soc. 1916, 23, 88-90) finds 

 an interesting theorem on the representability of a linear 

 function of a line where the function has continuity of the 

 nth. order. 



In 1 91 4 Fischer published a paper in which he defined the 

 derivative of a function of a surface and proved some im- 

 portant theorems about such derivatives. He now (Amer. 

 Journ. Math. 191 7, 39, 123-34) considers functions depending, 

 not only on the surface, but also on the values taken by a 

 function at every point of the surface : such a function has 

 two partial functional derivatives. Similar work for functions 

 of lines was published by Levy in 1914. 



Mary Evelyn Wells (ibid. 163-84) discusses the inequalities 

 found by E. H. Moore in the theory of the general linear integral 

 equation. These inequalities include an analogue of the in- 

 equality of Schwarz in the theory of Fredholm's linear integral 

 equations. 



Geometry. — In 191 2 E. Kasner published some results of his 

 study of the invariants of a pair of analytic curves under the 

 " equilong " groups with the main object of throwing light on 

 the corresponding question in the more important conformal 

 geometry. The two theories present many analogies, but are 

 not connected by a strict principle of duality. In some ques- 

 tions the two theories differ essentially both in the appropriate 

 methods and the results ; and in the paper mentioned the con- 

 vergence of the power series entering into the formal calcula- 

 tions was not proved. Recently Kasner (Bull. Amer. Math. 

 Soc. 191 7, 23, 341-7) completed the equilong theory by showing 



