334 SECTIONAL TRANSACTIONS.— A*, 



(i) First results on the infinitesimal structure of closed surfaces and the 

 theory of harmonic transformations of complexes. 



(2) Solution of the problem of intersections of algebraic complexes and 

 arbitrary closed sets. 



(3) Infinitesimal theory of general spaces and the principle of inductive 

 linkages. The new Pflaster theorems. 



(5) Solution of Alexandroff 's problem on homologies in the large. 

 In conclusion, some problems of general topology are discussed in the 

 light of the new theory. 



Dr. A. C. Offord. — Uniqueness theorems for trigonometric series and 

 integrals (12.10). 



Cantor's fundamental uniqueness theorem for trigonometric series 

 asserts that if 



lim ^ ,-„v 



Zi c n e lnx = o 



for all x in (o, 27i), then c n = o for all n. It is natural to suppose that there 

 is a strictly analogous theorem for trigonometric integrals, and this is in fact 

 true. Thus if cp(w) is integrable L in every finite interval and if 



(1) Jl^r 9(u)e^"du = o 



J — CO 



CO 



for all x, then <p(w) is equivalent to zero. 



It is to be observed that the integral in (1) may converge for all x however 

 great the order, or average order, of |<p(w)|. For example, it is convergent 

 for all x when 



<p(w) = exp (aw + ie" 2 ) o < a < 1 



Cantor's theorem has been generalised very widely by various writers, 

 in particular by Rajchmann and Zygmund and by Verblunsky. There are 

 corresponding results for trigonometric integrals and one of the most 

 interesting of these is the following theorem. 



If cp(«) is integrable L in every finite interval and if 



(-?) 



ep(w) e ixu d u = o 

 for all x, then <p(«) is equivalent to zero. 



Miss A. Cox.- — On representation by squares and quadratfrei integers in a 

 real quadratic corpus (12.40). 



Applications of the Hardy-Littlewood method to problems of additive 

 arithmetic in algebraic corpora have been made by Siegel and others. In 

 the present paper, which describes joint work of Dr. Linfoot and the 

 author, Siegel's arguments {Math. Annalen, 87 (1922), 1-35) are applied 

 to the problem of the representation of integers of large norm in a real 

 quadratic corpus k(\/d) as a sum of squares and total-positive quadratfrei 

 integers of the corpus, a quadratfrei integer of a corpus being defined as 

 one not divisible by the square of a prime ideal of the corpus. 



