a$_{21}$m ( s ) V ′ ( t$_{0}$ ) = − λ ( s ) sinh θ, a$_{31}$n ( s ) W ′ ( t$_{0}$ ) = − λ ( s ) cosh θ, U ( t$_{0}$ ) = V ( t$_{0}$ ) = W ( t$_{0}$ ) = 0 θ ( s ) = − s ∫ s 0 τds + θ$_{0,}$ (4.5) (ii) If we choose Example 7 Let r ( s ) = (cos ( s ) , sin ( s ) , 0) be an arc-length spacelike curve 0 < s ≤ 2 π. It is easy to show that λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C 1 functions, a$_{ij}$ ∈ R ( k = 1 , 2 , 3; j = 1 , 2 , 3 , ..., p ). U ( t$_{0}$ ) = V ( t$_{0}$ ) = W ( t$_{0}$ ) = 0 and f (0) = g (0) = h (0) = 0 , θ ( s ) = − s ∫ s 0 τds + θ$_{0}$, (4.6) g ′ (0) a$_{21}$m ( s ) V ′ ( t$_{0}$ ) = − λ ( s ) sinh θ, h ′ (0) a$_{31}$n ( s ) W ′ ( t$_{0}$ ) = − λ ( s ) cosh θ, λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C 1 functions, a$_{ij}$ ∈ R ( k = 1 , 2 , 3; j = 1 , 2 , 3 , ..., p ). u ( s, t ) = f ( p ∑ k =1 a$_{1}$$_{k}$l ( s ) k U ( t ) k ) , v ( s, t ) = g ( p ∑ k =1 a$_{2}$$_{k}$m ( s ) k V ( t ) k ) and w ( s, t ) = h ( p ∑ k =1 a$_{3}$$_{k}$n ( s ) k W ( t ) k ) then, we can express the sufficient condition for which the curve r ( s ) is a line of curvature on the surface P ( s, t ) as T ( s ) = ( − sin ( s ) , cos ( s ) , 0) , N ( s ) = ( − cos ( s ) , − sin ( s ) , 0) , 15