SECTIONAL TRANSACTIONS.—A*, 345 
motion of a general system can be reduced to that of a particle in a hyperspace. This 
is especially of value in the discussion of the stability of a dynamical system, which 
becomes the geometrical problem of the variation of certain curves in a representative 
space. In particular, we can choose the space so that the dynamical trajectories 
are represented by geodesics and the problem of stability becomes identical with that 
of geodesic deviation in differential geometry. 
The study of null-geodesics of a Riemannian manifold is of some importance since 
they are the paths of light-rays in the space-time of relativity theory. It is possible 
to choose a space in which the trajectories of a dynamical system are represented by 
its null-geodesics. Such a representation leads to a geometrical theory of the 
Hamilton-Jacobi equation and shows the intimate connection between the geometrical 
and dynamical theories of contact transformations. 
Dynamical systems are of two types, holonomie and non-holonomic, and our 
remarks so far have been confined to the first. The attempt to find a geometrical 
representation of non-holonomic systems has led to a new type of differential geometry, 
which treats of the properties of non-holonomic manifolds. Here again we can define 
the geodesics of such a manifold and showthat they represent the trajectories of the 
corresponding dynamical system. 
Prof. J. B. 8. Hatpane.—Some Mathematical Problems of the Biologist. 
In a population in which characters are inherited according to Mendel’s laws, 
and which is subject to an assigned mating system and type of selection, it is 
required, given the population in one generation, to predict its composition in 
future generations. If the population be large and generations do not overlap, the 
parameters defining the composition of the (x + 1)th generation can be given in terms 
of those of the nth. Thus, in the trivial case of self-fertilization the proportion 7, of 
heterozygotes is given by the equation p,,,;=4,, .°. p, = (4)"p,. If all matings are 
between (m— 2)th cousins p, is the sum of m geometric series whose common 
tatios are the roots other than 4, of a+! — ym+2-m-1 = 0. More complex problems 
lead to groups of as many as 22 simultaneous linear finite difference equations. 
Tf selection occurs we have to solve non-linear difference equations of which 
2 
Aw, — 138 typical. When kf is small they can be solved as differential equations, 
n 
and the effect of natural selection on a population whose variation is due to m genes 
can be represented by the motion of a point in m-dimensional space. The trajec- 
tories fall into families converging to from | to 2”~! stable equilibria. In general 
the equation Aw, = ke(u,,) may be solved by expanding in ascending powers of k. 
The convergence of the series obtained is under investigation. 
Monday, September 28. 
Discussion on Fluid Motion with reference to Aerodynamics :— 
Prof. G. I. Taytor, F.R.S.—Stability and Turbulence in a Stream of 
Fluid of Variable Density. 
Stability of Waves.—The waves in « stream of fluid, the density and velocity of 
which vary with height above a horizontal plane, are investigated mathematically. 
Tn the case of a fluid moving with uniform shearing velocity U=«z parallel to the axis 
of x and possessing a small uniform rate of decrease in density with height expressed 
by B As s e being the density and z the height, it is found that if «? > 4g no 
p 
_ waves are possible either stable or unstable ; on the other hand, if « < 4 gf, all waves 
appear to be stable. 
Other cases are investigated where the fluid of variable density is replaced by 
_ superposed layers of fluid each of constant density, but each less dense than the 
fluid below it. The velocity distribution is the same as before, namely, a uniform 
