A self-adjoint positive-definite variational principle is presented which leads to upper and lower bounds for < S*, ϕ >, where < S*, ϕ > is an integral over position and angular direction of the product of the one-velocity neutron transport flux, ϕ and an arbitrary adjoint source, S*. The Euler equation of the functional is a new form of the one-velocity Boltzmann neutron transport equation in which the dependent variable is one-half the sum of ϕ and ϕ*, where ϕ* is the adjoint flux. When a trial function consisting of an expansion in spherical harmonics is used, one obtains as a lower bound for < S*, ϕ > the quantity < US1, ϕ(P−N′; S1) > − < US2, ϕ(P−N″; S2) >, where S1(r, Ω) = [S(r, Ω) + S*(r, −Ω)]/2, S2(r, Ω) = [S(r, Ω) − S*(r, −Ω)]/2, ϕ(P-N′; S1) is an odd P−N approximation to a problem with the same cross sections as the original problem, but with source S1; ϕ(P−N″; S2) is an even P−N approximation to a problem with source S2, and U is the operator that takes a function f(r, Ω) into f(r, −Ω).