Secondly, since S ( T ) = ωT, ω = 0 , where s$_{0}$ is the starting value of arc length and θ = θ ( s ) . In this paper, we assume that s$_{0}$ = 0 . Combining (4.8), (4.9) and (4.10), we have the following theorem. Theorem 8 A timelike curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if the followings are satisfied We call the set of surfaces defined by (4.1) - (4.4) or (4-7) - (4-10) timelike surface pencil with a common line of curvature . Any surface P ( s, t ) satisfying these conditions is a member of this family. Now, we analyse two different types of the marching-scale functions 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 (i) If we choose marching-scale functions as u ( s, t ) , v ( s, t ) and w ( s, t ) in the Eq. (4.1). 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}$n ( s ) k W ( t ) k θ ( s ) = s ∫ s 0 τ ds + θ (0) , u ( s, t$_{0}$ ) = v ( s, t$_{0}$ ) = w ( s, t$_{0}$ ) ≡ 0 , φ$_{1}$ ( s, t$_{0}$ ) ≡ 0 , φ$_{2}$ ( s, t$_{0}$ ) = λ ( s ) cos θ, φ$_{3}$ ( s, t$_{0}$ ) = λ ( s ) sin θ. θ ( s ) = s ∫ s 0 τds + θ$_{0,}$ (4.10) 19