B S$_{1}$ i ( U, J $^{π}$) B S$_{2}$ i ( U, J $^{π}$) = R n$_{1}$ i ( U ) R n$_{2}$ i ( U ) (4) R n$_{j}$ i ≡ σ n$_{j}$ i σ n$_{j}$ R . (5) R x$_{j}$ i ( U ) ≡ ∑$_{J}$ π σ $^{x$_{j}$}$( U, J $^{π}$) · B x$_{j}$ i ( U, J $^{π}$) ∑$_{J}$ π σ $^{x$_{j}$}$( U, J $^{π}$) , (6) B S$_{1}$ i ( U, J $^{π}$) = B S$_{2}$ i ( U, J $^{π}$) · R n$_{1}$ i ( U ) R n$_{2}$ i ( U ) . (7) R S$_{1}$ i ( U ) = ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) · B S$_{1}$ i ( U, J $^{π}$) ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) = ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) · B S$_{2}$ i ( U, J $^{π}$) · R n 1 i ( U ) R n 2 i ( U ) ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) = R n$_{1}$ i ( U ) R n$_{2}$ i ( U ) · ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) · B S$_{2}$ i ( U, J $^{π}$) ∑$_{J}$ π σ $^{S$_{1}$}$( U, J $^{π}$) . (8) R S$_{1}$ i ( U ) = R n$_{1}$ i ( U ) R n$_{2}$ i ( U ) · ∑$_{J}$ π ασ $^{S$_{2}$}$( U, J $^{π}$) · B S$_{2}$ i ( U, J $^{π}$) ∑$_{J}$ π ασ $^{S$_{2}$}$( U, J $^{π}$) = R n$_{1}$ i ( U ) R n$_{2}$ i ( U ) · R S$_{2}$ i ( U ) (9) 7 by definition. This equation is equivalent to Eq. (2). Since we know R n$_{2}$ i ( U ), and we measure R S$_{1}$ i /R S$_{2}$ i in surrogate ratio method, we can obtain R n$_{1}$ i to the accuracy mentioned above. This gives an explanation of the reason why the surrogate ratio method works. Since the 2 surrogate reactions S$_{1}$ and S$_{2}$ are assumed to be carried out for a pair of nuclei having similar mass and structure, the distribution of the formation cross section σ $^{S$_{1}$}$( U, J $^{π}$) will be fairly close to that of σ $^{S$_{2}$}$( U, J $^{π}$) if the nuclear structure and reaction mechanisms are similar to each other. We can write this similarity as σ $^{S$_{1}$}$( U, J $^{π}$) = ασ $^{S$_{2}$}$( U, J $^{π}$), where the symbol α denotes a constant such as the kinematical factor. If the dependence of α on J π is ignorable, Eq. (8) reads Then, the decay probability for reaction i in surrogate S$_{1}$ measurement, R S$_{1}$ i , can be written as where σ $^{x$_{j}$}$( U, J $^{π}$) denotes the formation cross section of J π states in reaction x$_{j}$ including the factor of (2 J + 1). Equation (4) can be rewritten as Relation of the B x i and R x i ( x = S$_{j}$ or n$_{j}$ ) are expressed as follows: to the accuracy mentioned above, where Figs. 3 and 4 can be expressed as