SECTIONAL TRANSACTIONS.—A*. 457 
perties, when considered according to the general theory of quadratic 
differential forms. 
It can be shown that the anti-symmetric tensors B(i, . . . ip) which satisfy 
the equations 
prt . . . . . 
D> ( = 1)r-t Bay, + + 6 lp-rly+1 . .. ptr, Zr) =o 
r7=1 
PIBGNs 24. By 1% 5) S10 
and which are finite everywhere on the manifold are all linear combinations 
with constant coefficients of Rp independent tensors, where Ry» is the p-th 
Betti number of the manifold, and that the integrals formed from these 
| Be, es tp) dx... dx'e, 
which are called harmonic integrals, cannot be without periods. It is usual 
to take 21, . . . , 2” as the complex parameters on the variety, and writing 
at = or inet, 
we take x1, ..., x?” as the real parameters on the manifold. Then 
among the harmonic integrals are included the real and imaginary parts of 
the Abelian integrals (of the first kind) 
| PG 2. tp) dz... dz'p 
attached to the variety. A study of the harmonic integrals leads to many 
new and interesting properties of the Abelian integrals, some of which are 
described. 
Dr. D. W. Bassace.—Cremona transformations (11.30). 
If V; is a rational k-dimensional locus in space S,, of m dimensions, 
which can be birationally projected from each of two [n — k — 1]’s, Il, and 
II,, then we can use V;, to set up a Cremona (1, 1) correspondence between 
two [R]’s, S,% and S,, taken in general position in S,, two points, 
P,, Ps, of these spaces corresponding when the [7 — k]’s, which join II, to 
P, and Il, to P, respectively, meet V, in the same point P. Segre has 
obtained Cremona transformations of ordinary space arising by two pro- 
jections in this way from rational scrolls of planes, and Marletta has given 
a simple method by which any Cremona transformation T can be inter- 
preted in terms of two projections of a locus of higher space; but apart 
from the work of these, little has been done by hyperspatial methods. 
In the present paper these methods are used to give a simple interpretation 
and classification of the so-called rational and elliptic Cremona trans- 
formations of ordinary space S3, the genus of a Cremona transformation 
of S; being defined with Loria as the genus of the general plane section 
of a general member of one of the two homaloidal systems of the trans- 
formation, a number which is an invariant of the transformation. The 
question of resolving a Cremona transformation T into the product of several 
simpler transformations is often simplified when T is given a hyperspatial 
interpretation ; for example, the known fact that all the rational Cremona 
transformations of 3 can be built up from quadro-quadric Cremona trans- 
formations is rendered practically self-evident. 
Dr. P. Du Vat.— Multiple planes (11.50). 
A multiple plane of n sheets may be defined as the projective image of 
a rational involution of sets of m points on an algebraic surface. It has a 
R 
