308 SECTIONAL TRANSACTIONS.— A, B. 



given of recent work in this connection, and it is shown how Riemann's relations 

 between the periods of Abelian integrals and the inequalities satisfied by the real and 

 imaginary parts of the periods can be generalised to apply to Abelian integrals of any 

 multiplicity. 



Mr. L. C. Young. — Continuous Groups and the Foundations of Geometry. 



Dr. H. W. Richmond. — A problem on Cubes of Rational Numbers. 



The object of this paper is to call attention to some work, published half a century 

 ago and apparently overlooked, upon an arithmetical problem closely allied to 

 problems which have recently been discussed at considerable length. With a few 

 unimportant exceptions, all rational numbers are expressible as a sum of cubes of 

 two rational numbers either in an infinity of ways or not at all. Sylvester and Pepin 

 discovered many types of numbers of the latter class ; but of rules for recognising 

 numbers of the former class or of methods for resolving them into a sum of cubes, 

 very little is known. Sylvester, however, made the statement that he had succeeded 

 in resolving all the whole numbers up to 100, other than those excluded by his rules, 

 with the single exception of 66. He did not publish his results ; and so the problem 

 has been left. 



Prof. P. J. Daniell. — The Mathematical Theory of Flame Motion. 



A new formula for the speed of propagation of flame into unbumt gas is obtained 

 by considering the rate of variation of combustion with temperature. The resulting 

 differential equation leads, after somewhat intricate approximations, to a formula 



y2=k (To/T) S2 exp (-E/RT) 



where S is the speed of sound at the initial temperature Tq, while T is the final tem- 

 perature ; ifc is a pure constant depending mainly on the proportions of the mixture. 



SECTION B.— CHEMISTRY. 



Thursday, September 4. 



Presidential Address by Prof. G. T. Morgan, O.B.E., F.R.S., on 



A State Experiment in Chemical Research. (See p. 38.) 



Prof. Sir William Pope, K.B.E., F.R.S., and Dr. J. B. Whitworth.— 

 The Resolution into Optically Active Components of the Spiro-5:b- 

 Dihydantoin. 



Several spiranes of enantiomorphous molecular configuration which contain no 

 asymmetric carbon atom in the molecule have been resolved into their optically active 

 components ; the spiranes thus hitherto studied have been of considerable molecular 

 complexity. 



It seemed to us desirable to attempt the resolution of an enantiomorphous spirane 

 of very simple constitution in the hope that the examination of simple substitution 

 derivatives of such optically active substances would yield further information con- 

 cerning the stability of molecular complexes. For our purpose we chose the 

 spiro-5:5-dihydantoin 



NH-CO NH-CO 



\/ 



C 



/ \ 

 CO-NH CO-NH, 



described by Biltz and his pupils in 1917. On fractional crystallisation of the brucine 

 salt of the synthetic material from water, Z-brucine-Z-spiro-5:5-dihydantoin is readily 

 obtained in clusters of thin, colourless needles melting at 260° ; from this the Isevo- 

 spirodihydantoin is separated in the usual way. 



The specific rotatory power of Z-spiro-5:5-dihydantoin for the mercury green line 

 is, in ethyl alcohol solution, [a] = — 113° and, in aqueous solution, [a] = — 115°. 



