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SECTIONAL TRANSACTIONS.—A*. 349 
identities, and the determination of these in large classes of algebras is a subject 
which remains for investigation. 
Some mention must also be made of work bearing on the infinite (quantum) 
algebras associated with the singular linear equationrp — pr= 1, which has nosolution 
in any associative finite algebra. So far, little interest has been shown in the more 
general problem of finding algebras in which some other given singular equation has 
a solution. 
Finally, non-commutative algebra has also been applied to the theory of functions 
and of partial differential equations although, strangely enough, not by the aid of 
continuous groups. 
Mr. T. Smiru.—Tesseral Matrices. 
A matrix wu and its reciprocal U, are tesseral matrices if 
tue fia O as c—a 
w= °) , ee | ») 
a, 6, c, d being similar matrices. Subject to the condition of rank, tesseral matrices 
such as 
ea eaf—A 
a Ghee alr 
where A is the reciprocal of a, and e and f are symmetrical, form a group. The multi- 
plication of matrices of this type, with a simple substitution, yields a solution of the 
problem of evaluating the function h 
h (x, z)=f (x, y)+9(y 2), 
where f and g are known algebraic functions of the sets of variables 2, y, z, when 
f+ q is stationary for any small changes in y. 
Dr. J. Wisnarr.—Interpolation without Differences. 
The common interpolation formule involve the differences of the tabulated 
function, and it has hitherto been considered desirable to furnish a table with, at any 
rate, the even differences in order to lessen the burden to the user. C. Jordan gave in 
1928 a formula of such a nature that tabular differences are of no value in facilitating 
its use. The general adoption of this formula would therefore much reduce the 
labour and expense of preparing new tables, provided it were clearly proved that the 
new method was an improvement, in the nature of the operations and the time involved, 
over the double operation of (a) calculating the differences, and (b) performing the 
interpolation by one of the ordinary methods. This question is discussed in the 
present paper. 
Dr. G. TempLe.—The Relativistic Wave Equation. 
The wave equation for a free electron has been constructed by Dirac by rationalisa- 
tion of the relativistic Hamiltonian equation, by Darwin and Frenkel on the analogy 
of Maxwell’s electromagnetic equations, and by Eddington from a general transforma- 
tion theory. 
The equation for one electron in an electromagnetic field gives an immediate 
explanation of the duplexity phenomena in optical spectra in terms of the ‘spin’ 
of the electron; but, although this result is one of the great successes of Dirac’s 
theory, there seems to be no satisfactory theoretical basis for the way in which the 
electromagnetic potentials are introduced. 
Numerous attempts have been made to construct a theory of the direct interaction 
of electric changes. Gaunt’s theory does not satisfy the requirements of relativity ; 
Eddington’s theory is in marked disagreement with experiment; Breit and other 
American physicists have tested, without success, various expressions for the inter- 
action energy. 
The only hope seems now to lie in developments of the quantum theory of electro- 
magnetic wave fields. 
