We call the set of surfaces defined by (3.1) - (3.4) spacelike surface pencil with a common line of curvature . Any surface P ( s, t ) defined by (3.1) and satisfying (3.2) - (3.4) is a member of this family. Now, we analyse two different types of the marching-scale functions u ( s, t ) , v ( s, t ) and w ( s, t ) in. the Eq. (3.1). then, we can simply express the sufficient condition for which the curve r ( s ) is a line of curvature of the surface P ( s, t ) as (ii) If we choose (i) If we choose 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 9 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}$$_{k}$n ( s ) k W ( t ) k ) λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C 1 functions, a$_{ij}$ ∈ R ( k = 1 , 2 , 3; j = 1 , 2 , 3 , ..., p ). a$_{21}$m ( s ) V ′ ( t$_{0}$ ) = λ ( s ) sinh θ, a$_{31}$n ( s ) W ′ ( t$_{0}$ ) = λ ( s ) cosh θ, U ( t$_{0}$ ) = V ( t$_{0}$ ) = W ( t$_{0}$ ) = 0 θ ( s ) = − s ∫ s 0 τds + θ$_{0,}$ (3.5) u ( s, t ) = p ∑ k =1 a$_{1}$$_{k}$l ( s ) k U ( t ) k , v ( s, t ) = p ∑ k =1 a$_{2}$$_{k}$m ( s ) k V ( t ) k and w ( s, t ) = p ∑ k =1 a$_{3}$$_{k}$n ( s ) k W ( t ) k u ( s, t$_{0}$ ) = v ( s, t$_{0}$ ) = w ( s, t$_{0}$ ) ≡ 0 , φ$_{1}$ ( s, t$_{0}$ ) ≡ 0 , φ$_{2}$ ( s, t$_{0}$ ) = λ ( s ) cosh θ, φ$_{3}$ ( s, t$_{0}$ ) = λ ( s ) sinh θ.