338 . SECTIONAL TRANSACTIONS.— A. 



coefficient of compressibility and elasticity has also been calculated and found to 

 be in fair agreement with the observed values (where these exist). 



The information about ionic and atomic forces which has been obtained has been 

 applied to other problems of crystal structure as yet unsolved. It has, for instance, 

 thrown considerable light on the question as to why certain crystals such as CsCl 

 set as body-centred cubics, while others such as NaCl set as face- centred cubic8. 



3. Sir William Bragg, K.B.E., F.R.S. — Demonstration of Recent 



Crystal Models. 



4. Sir E. Rutherford, O.M., F.R.S., and Dr. J. Chadwick. — Collision 



of a Particles with Light Atoms. 



5. Mr. W. W. Garrett. — On Transformation of Elements by Low Voltage 



Discharges. 



6. Dr. F. J. M. Stratton. — -The Recent British Eclipse Expedition to 



Sumatra. 



7. Prof. H. H. Turner, F.R.S.— Owr Coming Total Eclipse.'] 



Department of Mathematics. 



8. (a) Mr. E. C. Francis. — The Evolution of the Concept of an Integral from 



Riemann to Stieltjes. 



(b) Prof. C. Caratheodory. — Some Applications of the Lehesgue 



Integral in Geometry. 



Though most people think that the Lebesgue theory of measure (and, what is 

 practically the same, his theory of integration) has been only devised for the purpose 

 of the theory of functions of a real variable, there are many instances where problems 

 of geometry, ordinary analysis, or even mathematical physics, cannot be dealt with, 

 if one discards the modern theories of sets of points and of measurement. The 

 following three examples illustrate this fact : — 



1. One of the simplest problems of geometry is to draw a tangent to a curve, 

 and the simplest curves are those of finite length. The Lebesgue theory shows that 

 those last curves have ' nearly everywhere ' a tangent; that is, that if you take a point 

 at random on the curve (or Ijetter, on an arc of the curve which you have rectified), 

 there is the probability one that the curve will have a definite tangent at the given 

 point. 



2. The simplest analytical functions are those that are regular and bounded 

 throughout the interior of the unit circle. If j'ou take a point at random on the circle 

 itself there is the probability one that the limit line /(re 5), as z-j^l, does exist. 



3. The third example is the celebrated theorem of Poincare that there is the 

 probability one that in a steady motion of an incompressible liquid the path of a given 

 molecule does return to any neighbourhood of the place where the molecule was located 

 at the time t= 0. At the time Poincare gave this theorem (1890) it was impossible to 

 understand the meaning of it, and Poincare's proof was of course inaccurate. But 

 twelve years later Borel and Lebesgue invented the new theory of measure by which 

 the very proof of Poincare was salved. It is now possible to give a much shorter proof 

 of Poincare's theorem of return that the original one.^ 



(c) Prof. G. H. Hardy, ^ .^.^. ^Trigonometrical Series. 

 A survey of the subject. 



1 See C. Caratheodory, ' Uber den Widerkehrsatz von Poincare ' (Sitzber. Berl. 

 Akad. 1919, p. 580). 



