656 
MATHEMATICS: G. D. BIRKHOFF 
Any one of the transformations K~^Tx k effects a translation on the 
three sides, AB, CD, DA of and carries the side BC into the curve to 
which T2F1 carries the linear segment in which the square is met by a 
line parallel to AD at a distance x from AD. Since if Tx is the transla- 
tion which has the same effect as A~^TxA on the point A , the transforma- 
tion Tx~^A~^TxA leaves all points of the three edges AB, CD, DA of 
Si invariant. Denote Tx~^A~^TxA by Fx. The set of transformations 
Fx (0 S X ^1) is obviously a continuous one-parameter family of 
(1 — 1) continuous transformations; i^o is the identity; and Fi the given 
transformation already denoted by Fi. Hence Fi is a deformation. 
The last paragraph can be replaced by the observation that since the 
product of two deformations is a deformation, Fi, which is the product 
of T2~^ and T2F1, must be a deformation. It seems worthwhile, however, 
to indicate, as has been done, something of the nature of the family of 
transformations Fx which the process sets up. 
A THEOREM ON SERIES OF ORTHOGONAL FUNCTIONS WITH 
AN APPLICATION TO STURM-LIOUVILLE SERIES 
By George D. Birkhoff 
1. The Theorem. — An infinite set of continuous functions Ui{x), thix), 
... is closed on the interval 0 ^ a; ^ 1 if there exists no continuous 
function f (x) not identically zero for which J^lf (x) u„ (x) dx vanishes for 
all n; the set is normalized if (^^ = ^ ^-ll n; it is orthogonal if 
J^l Um (x) Un {x)dx = 0 for w ^ n. Most of the series of mathematical 
physics are linear in closed normalized orthogonal sets of functions. 
Tttf.orf.m. // Ui{x),U2(x), . . . form a closed normalized orthogonal set 
of functions, and ifui{x), ii^ix), . . . form a second normalized orthogonal 
set such that 
converges to a function H {x, y) less than 1 in numerical magnitude in such 
wise that the series multiplied through hy an arbitrary continuous function 
f(x) can be integrated term by term as to x and yields a uniformly conver- 
gent series, then the set Ui{x), U2{x), . . . is closed also. 
Proof. If the set Ui{x), U2{oo), . . . is not closed there exists an / 
not identically zero such that J^lf {x)u„{x) dx vanishes for all n. In 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Communicated by E. H. Moore, October 10, 1917 
