348 SECTIONAL TRANSACTIONS.—A*. 
definite transcendental equation which A and uw together must satisfy, and there 
are certain others which they must not satisfy. The simplest case is that of 
linear velocity, ¥o = —Uy?/2H, which includes the case of fluid initially at rest; 
for this particular case the actual transcendental equations will be given. 
The study of periodic disturbances of finite amplitude is rendered difficult by 
the existence of small and zero divisors. The series for ¥ (2, y, ¢) is real only when 
¥7- has the form 
k 
> ox [— {aX +e — Wa} w— fat —AMul t |. fen ny), 
jit 0) 
when A, A; u,p: andfi—n,n (y)s fr. k-n (y), are conjugate complex numbers. 
Prof. J. R. Partineron and Mr, N. L. Anrrtocorr.—Curved Prpe Stream 
Tane Flow and Viscosity. 
Viscosity measurements by means of transpiration through a helical capillary are 
quite usual, but recent determinations by this method into the high temperature 
region have given rise to unexpected anomalies in the Sutherland Law. 
Tt is now shown that these latter were due partly to curved pipe stream line 
motion at the lower temperatures and partly to a probable real breakdown in 
Sutherland’s Law at high temperatures. Corrections by means of White’s method 
of plotting Dean’s Criterion have been found to be applicable, and the Dean formula 
to hold within the limits of experimental accuracy for low values of increased 
resistance for Argon, Air and Hydrogen Chloride. 
It has further been demonstrated by means of Dean’s Criterion that inconsistencies 
in some of the early viscosity determinations are due to curved pipe flow and hence 
the explanations appearing in the literature for these are probably incorrect. The 
low temperature determinations of the Halle School are in order, with the exception 
of the lowest temperature determination on Argon, the latter result being one of the 
range employed by Lennard Jones and by Hassé and Cook in their theories of the 
variation of viscosity with temperature. 
GENERAL Discussion (Sir Horace Lamp, F.R.8.). 
Wednesday, September 30. 
Prof. A. R. Ricuarpson.—Recent Developments of Non-commutative 
Algebra. 
During the last ten years no fewer than 100 papers have been published dealing 
with different aspects of non-commutative algebra. 
Of these, many are concerned with division algebras, that is, number systems 
in which given any non-zero number 2, y can be found, such that zy =1, the 
investigation of which was rendered fruitful by L. E. Dickson’s definition of an 
infinity of such algebras by the aid of algebraic equations. Work concerning 
their structure has failed, however, to determine whether ali division algebras can 
be so constructed from cyclic equations. 
Progress in other branches of the subject has been made possible by the definition 
of integers in non-commutative systems. For example, in every algebra there is an 
equivalent of Fermats’ theorem, and the theory of congruences, diophantine equations, 
rings and modules is being studied and applied to real integral number theory, in 
particular to the solution of diophantine equations. Work remains to be done on the 
theory of quadratic reciprocity and, notwithstanding the introduction of ideals, the 
problem of restoring the uniqueness of factorization is still unsolved. 
© The definition of determinants in non-commutative algebras has assisted the study 
of the theory of linear substitutions and invariants in such systems. While, however, 
a solution has now been found for the general linear equation, the quadratic, except 
in the very simplest cases, must still be regarded as unsolved. One of the difficulties 
presented by such equations is due to the existence of many linear and quadratic 
