TRANSACTIONS OF SECTION A. 533 



sphere. If two spheres lying in the same space of three dimensions touch, then 

 the line joining a focus of one to one of the foci of the other is an isotropic 

 line— i.e., a line of zero length. 



By considering the spheres which have the same kind of contact with two 

 others and making use of the fact that the four points of contact lie on a circle 

 we may deduce the following theorem : If the sides of a skew quadrilateral are 

 isotropic lines they lie on a sphere. 



In the same way we may prove that if the sides of skew hexagon in a space 

 of five dimensions are isotropic lines, they lie on a spherical manifold of four 

 dimensions. There is a similar theorem for a polygon of 2n sides in a space 

 of 2n — 1 dimensions. 



10. On the Theory of the Ideals. By Professor J. C. Field. 



In the elements of the theory of the algebraic functions it is shown that the 

 branches of an algebraic function r defined by an equation F (z, v)=0 as repre- 

 sented by a number of power-series in z — a in the neighbourhood of a value 

 z = a. These power-series have exponents integral or fractional, and group 

 themselves into a number of cycles, the branches of these cycles having certain 

 orders of coincidence with a given rational function Ii(z,v). In his book on 

 the algebraic functions the writer has defined these orders of coincidence as 

 adjoint when they are equal to or greater than certain numbers 



v, v r 



respectively, and has shown how to construct a rational function possessing any 

 assigned set of adjoint orders of coincidence corresponding to the value of the 

 variable in question. Hensel, in hifi book, ' Theorie der algebraischen Zahlen,' 

 and also in two earlier Abhandlungen, which appeared in Crelle's Journal (Bde. 

 127, 128), has shown that an ordinary algebraic equation / (x)=0 is satisfied 

 by certain series analogous to the well-known power-series referred to above. 

 These series have reference to a prime p, and also group themselves in cycles 

 like the more familiar series of the algbraic functions. On these series Hensel 

 builds up a theory of the ideals. For a more precise description of the series 

 here in question the reader is referred to the writings just cited. 



In the present paper the writer, on starting out from Hensel's power-series, 

 defines adjointness relatively to a prime p in a manner analogous to that in 

 which he defines the property in connection with the algebraic functions. 

 Employing the letter e to designate a solution of our algebraic equation, it is 

 shown that we can construct a rational function R(-e) possessing any assigned 

 set of adjoint orders of coincidence corresponding to a prime p. From this 

 it readily follows that we can construct a rational function possessing any set 

 of orders of coincidence with the branches of the cycles corresponding to the 

 prime in question. We can then construct a general function R(e) which 

 represents only integral algebraic numbers and which possesses a single coinci- 

 dence with the branches of an assigned one of the cycles corresponding to any 

 prime, while it is not conditional with regard to any other specific prime or 

 any other cycle corresponding to the prime in question. The aggregate of 

 numbers represented by this function is a prime ideal. 



11. Report on Bessel Functions. — See Keports, p. 37. 



MONDAY, SEPTEMBER 5. 

 The following Papers were read : — 



1. Demonstration of Vacuum-tight Seals between Iron and Glass. 

 By Henry J. S. Sand, Ph.D., D.Sc. 



Some cathode-ray tubes were shown in which a vacuum-tight seal between 

 iron and glass has been obtained as follows : An iron wire is sealed into a glasr 



