456 SECTIONAL TRANSACTIONS.—A*. 
When this is the case we say that F(x) belongs to the class B* and we take 
4(x) to be its cosine transform. 
We can now state our results as follows : 
(i) A function of H has a cosine transform in ‘De. 
(ii), A function of B* has a cosine transform in H. 
Suppose now that f(x) belongs to’ H and is bounded. Then we show that 
its transform F(x) belongs to H. Hence there is a class HB, consisting of 
all the bounded functions which belong to H, which is such that the cosine 
transform of a function of HB belongs also to HB. 
These results hold also for Hankel transforms. We say that f(x) belongs 
to Hy if 
Ww 
[f(s =) won syle) fd de | <M 
° wW 
for allx and w. There is a corresponding definition of the class By*, and 
there is a symmetrical class HyB as in the case of the cosine transform. 
We can now apply the analysis of Hardy and Titchmarsh (Quart. Fournal, 
Ox. Series, i, pp. 196-231) to find the class of all the functions of HyB 
which are their own Hankel transforms. We obtain, in fact, the following 
result. 
A necessary and sufficient condition that a function f(x) of HyB should be 
its own Hankel transform is that it should be of the form 
co 
I ’ 
f(s) = = x(t) 2 -4-i di, 
—co 
where the integral is summable (C, 1) almost everywhere, and x(2) is such 
that 
io 
la m8 
| [6 laa) x(t) x dt| < M, 
x(t) ti finctl 
: = tion of t. 
st FRG eg ea a) even function o 
Putting v = — 4, we get the necessary and sufficient condition for a 
function to be its own cosine transform. 
It is possible to extend the theory for cosine transforms to functions of 
several variables. 
Friday, September 8. 
Mr. W. V. D. Hopce.— Abelian integrals attached to algebraic varieties (11.0). 
G. Mannoury has shown how a complex projective plane can be repre- 
sented as a closed four-dimensional locus in Euclidean space in such a way 
that many projective properties of the plane can be represented as metrical 
properties of the locus. His method can be extended to algebraic varieties 
of any number of dimensions, and in this way we can obtain a representation 
of the Riemannian manifold of an algebraic variety of m dimensions as a 
locus of 2m dimensions in a Euclidean space. The differential form 
which gives the element of length on this locus has many interesting pro- 
