U ( t$_{0}$ ) = V ( t$_{0}$ ) = W ( t$_{0}$ ) = 0 and f (0) = g (0) = h (0) , s θ ( s ) = − ∫ s 0 τds + θ$_{0,}$ (3.6) 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 ). 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, Example 4 Let r ( s ) = ( a sinh ( s c ) , bs c , a cosh ( s c )) be a spacelike curve , a, b, c ∈ R , a 2 + b 2 = c 2 and − 2 ≤ s ≤ 2 . It is easy to show that T ( s ) = ( a c cosh ( s c ) , b $_{c}$, a c sinh ( s c )) , N ( s ) = sinh ( s c ) , 0 , cosh ( s c )) , B ( s ) = ( b c cosh ( s c ) , − a $_{c}$, b c sinh ( s c )) . By taking θ (0) = 0 we have θ ( s ) = − bs c 2 . If we choose λ ( s ) ≡ 1 , t$_{0}$ = 0 , a$_{21}$ = a$_{31}$ = 1 and u ( s, t ) = 3 ∑ k =1 a$_{1}$$_{k}$l ( s ) U ( t ) ≡ 0 , v ( s, t ) = sinh ( − bs c 2 ) t + 3 ∑ k =2 a$_{2}$$_{k}$ sinh k ( − bs c 2 ) t $^{k}$, w ( s, t ) = cosh ( − bs c 2 ) t + 3 ∑ k =2 a$_{3}$$_{k}$ cosh k ( − bs c 2 ) t k then the Eq. (3.5) is satisfied. Letting a = b = 1, we immediately obtain a member of the surface pencil (Fig. 3.1) as P$_{1}$ ( s, t ) = ( sinh ( s √$_{2}$ ) +sinh ( − s 2 ) t sinh ( s √$_{2}$ ) +sinh ( s √$_{2}$ ) 3 ∑ k =2 a$_{2}$$_{k}$ sinh k ( − s 2 ) t $^{k}$+ √$_{2}$ 2 cosh ( − s 2 ) t cosh ( s √$_{2}$ ) + √$_{2}$ 2 cosh( s √$_{2}$ ) 3 ∑ k =2 a$_{3}$$_{k}$ cosh k ( − s 2 ) t $^{k}$, s √$_{2}$ − √$_{2}$ 2 cosh ( − s 2 ) t − √$_{2}$ 2 3 ∑ k =2 a$_{3}$$_{k}$ cosh k ( − s 2 ) t $^{k}$, cosh( s √$_{2}$ )+sinh( − s $_{2}$) t cosh( s √$_{2}$ )+ cosh( s √$_{2}$ ) 3 ∑ k =2 a$_{2}$$_{k}$ sinh k ( − s 2 ) t k 10