458 SECTIONAL TRANSACTIONS.—A*. 
branch curve, locus of images of sets in which two points coincide, which 
has a cusp for each set in which three points coincide, and a node for each 
set in which four points coincide by pairs. The arithmetic genus of a 
multiple plane is given by 
Dart es (8. 1) (Remi2): the ex — 8 
where 7 is the number of sheets, 28 the order of the branch curve, 3x the 
number of its cusps, and 48 that of its nodes. 
Double planes form a class of surfaces somewhat analogous to hyper- 
elliptic curves, in having a rational involution of pairs of points; the 
analogous property holds, that the canonical system belongs to the 
involution. 
A double plane can have as branch curve any curve of even order, but for 
n > 2 the branch curve must have some cusps and (for m > 3) nodes in order 
that the surface may exist at all. 
A topological condition (of presence and arrangement of cusps and 
nodes) on the branch curve for the existence of a multiple plane exists but 
is not easy to apply. 
By comparatively simple algebraic methods, however, it is possible to 
enumerate all the cases that can arise with branch curves of reasonably 
low order. 
Mr. J. H. C. WuITEHEAD.—On the calculus of variations in the large: loct 
of conjugate points (12.10). 
Let Vn be an analytic manifold with a positive Finsler metric 
ds? = gij (x, dx) dx' dx, 
the g’s being homogeneous of degree zero. By minimising the integral 
fds we obtain a family of extremals. Each extremal through a given 
point O may be regarded as the image, possibly the singular image, of a 
straight line through a point (0) in a Euclidean space, En. As when 
setting up a normal co-ordinate system one can vary the straight line through 
(0) and so represent V» as the image of Ey in a single-valued analytic 
transformation E, > Vn. The points in Ey, at which this transformation 
fails to be locally (1-1) correspond to the points in Vn, which are conjugate 
to O. They constitute an analytic complex Ky... The object of this 
paper is to study the complex Ky, and the nature of the transformation 
En — Vn near points on Ky. 
Mr. H. G. Green.—Pascal’s Theorem in n dimensions (12.30). 
The paper describes the work of the author and a colleague on an appli- 
cation, which is still in progress, of the theory of involutions of restriction 
1 to a generalised Pascal figure. The methods used are a development 
from those of Pomey, which give opportunity for a closer discussion of 
special cases. The methods of the extension are illustrated by details of 
the figure in three dimensions, in which the place of the two dimensional 
Pascal line is taken by a series of closed networks of lines and the pro- 
jective connection with the plane figure is established. In the general 
n dimensional case it is shown that it is only for special forms of m that 
a symmetric figure can be constructed, and that the networks-are then of 
two types, open and closed. 
