SECTIONAL TRANSACTIONS— A*. 395 



delayed, and asymptotic series must be used. In straightforward applica- 

 tion, the accuracy is definitely limited by the size of the smallest term. 

 Two means of increasing the accuracy, the Euler transformation and the 

 convergence factor, are described. 



Recurrence formulas are very useful, but they usually lose accuracy when 

 used in one direction. This fact can upon occasion be turned to advantage. 



Two outstanding difficulties, with some slowly convergent series, and 

 with asymptotic series whose terms are all of the same sign, remain. 



Dr. J. C. P. Miller. — Step-by-step integration of a differential 

 equation, with some remarks on interpolation (12.0). 



Expansions in series, whether convergent or divergent, are often incon- 

 venient for the systematic tabulation of a solution of a differential equation 

 for some ranges of the argument, although they are indispensable as 

 checks. Hence, in order to compute pivotal values, i.e. values which form 

 a basis for subsequent subtabulation, we frequently use step-by-step pro- 

 cesses over some part of the range. Two such processes are briefly de- 

 scribed, namely the Double Summation and Taylor Series methods, as 

 applied to certain second order equations. 



Methods of checking and ways of estimating and minimising cumulative 

 errors, as well as the application of the Taylor Series method to interpolation, 

 are also considered. 



Dr. A. J. Thompson. — The printing of mathematical tables (12.30). 



The paper describes the several processes that come between the com- 

 pletion of the calculation of a mathematical table and its appearance as a 

 printed volume. Inter alia, it deals with the preparation of the printer's 

 copy, with typographic details (such as choice of type, spacing and rules) 

 and with the methods of ensuring accuracy. The standpoint is that of the 

 computer of the table, and technical matter is reduced to a minimum. 

 The paper is illustrated by photographs of a number of tables. 



Monday, August 22. 



Prof. G. D. BiRKHOFF. — Analytic deformations (lo.o). 



Prof. S. Lefschetz. — Fixed points of transformations (11. 15). 



The author first points out by examples the role of the problem in various 

 mathematical disciplines. If the elements transformed are points of 

 an abstract space R we are dealing with a problem in topology. 

 Suppose that we have a transformation T of R into itself. Under certain 

 very general conditions (C) if T is a transformation of R into itself there 

 may be given a topological character G (T) having the property that when 

 6 9^ o there is at least one fixed point. Conditions (C) embrace con- 

 tinuous single-valued transformations of a polyhedron into itself, and more 

 generally of a very broad class of locally connected spaces (absolute neigh- 

 bourhood retracts when R is compact metric). Special noteworthy case : 

 R is the Hilbert parallelotope. For all these cases an explicit expression of 

 e may be given in terms of the transformations which T induces on the 

 cycles of the space. 



