We want a static solution to the equations of motion, so it is convenient to set σ 0 = t and σ 1 = x , and assume y = z = 0. We also assume that the string is at a constant angular position in S $^{5}$. Then the action becomes with Equation (4.13) is very similar to the integral found when performing Fer- mat’s Principle for the path of light rays [102]. For large separation of endpoints, the trajectory which minimizes S locates itself very nearly at the minimum of n ( r ) for most of its length. Hence the complicated pro- cess of optimizing S is reduced to simply optimizing n ( r ). Furthermore, for small separations Δ x about its minimum, S becomes It is easy to show that n ( r ) is optimized by which gives with Σ = Δ − δ Δ+ $_{δ}$. (4.16) shows us that this geometry gives us a linear quark- antiquark potential. In other words, it exhibits confinement. 63 S = 1 2 πα ′ ∫ dt dx n ( r ) [ 1 + m ( r ) ( dr dx ) 2 ] 1 / 2 (4.13) n ( r ) = G$_{xx}$ = $_{H}$− 1 / $^{2}$f (Δ+ δ ) / 4 m ( r ) = $_{G}$− 1 $_{xx}$G$_{rr}$ = H w 4 − b 4 w 4 f (1 − δ +Δ) / 2 S ≃ n ( r$_{min}$ ) 2 πα ′ Δ x (4.14) r 4 min = 2 ω 4 ( Δ+ δ Δ − δ ) 1 /δ − 1 (4.15) V (Δ x ) = S (Δ x ) = Σ δ − Δ 4 δ 2 πα $_{′}$√$_{1}$ − Σ Δ x (4.16)