In his Lost Notebook, Ramanujan claimed that the “circular” summation of the
n
n
-th powers of the symmetric theta function
f
(
a
,
b
)
f(a,b)
satisfies a factorization of the form
f
(
a
,
b
)
F
n
(
a
b
)
f(a,b)F_{n}(ab)
. Moreover, Ramanujan recorded identities expressing
F
2
(
q
)
F_{2}(q)
,
F
3
(
q
)
F_{3}(q)
,
F
4
(
q
)
F_{4}(q)
,
F
5
(
q
)
F_{5}(q)
, and
F
7
(
q
)
F_{7}(q)
in terms of his theta functions
φ
(
q
)
\varphi (q)
,
ψ
(
q
)
\psi (q)
, and
f
(
−
q
)
f(-q)
. Ramanujan’s claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for
F
6
(
q
)
F_{6}(q)
,
F
8
(
q
)
F_{8}(q)
,
F
9
(
q
)
F_{9}(q)
, and
F
10
(
q
)
F_{10}(q)
.