Definition 2.1 Let α be a nonnull curve of osculating order 3. The harmonic functions
{H}_{j}:I\u27f6\mathbb{R},\phantom{\rule{1em}{0ex}}0\le j\le 1,
defined by
\{\begin{array}{c}{H}_{0}=0,\hfill \\ {H}_{1}=\frac{{k}_{1}}{{k}_{2}}\hfill \end{array}
are called harmonic curvatures of α, where {k}_{1}, {k}_{2} are Frenet curvatures of α which are not necessarily constant.
Definition 2.2 Let α be a timelike curve in {\mathbb{L}}^{3} with {\alpha}^{\mathrm{\prime}}(s)={V}_{1}. X\u03f5\chi ({\mathbb{L}}^{3}) being a constant unit vector field, if
\u3008{V}_{1},X\u3009=cosh\phi \phantom{\rule{1em}{0ex}}(\text{constant}),
then α is called a general helix (inclined curves) in {\mathbb{L}}^{3}, φ is called a slope angle and the space Sp\{X\} is called a slope axis [5].
Definition 2.3 Let α be a nonnull of osculating order 3. Then α is called a general helix of rank1 if
holds, where c\ne 0 is a real constant.
We have the following results.
Corollary 2.1

(i)
If {H}_{1}=0, then α is a straight line.

(ii)
If {H}_{1}is constant, then α is a general helix of rank1.
Proof By the use of the above definition, we obtain the proof. □
Proposition 2.1 Let α be a curve in {\mathbb{L}}^{3} of osculating order 3. Then
where {H}_{1} is harmonic curvature of α.
Proof By using the Frenet formulas and the definitions of harmonic curvatures, we get the result. □
Now, we will give the relation between harmonic curvature and a ccrcurve in {\mathbb{L}}^{3}.
Definition 2.4 A curve \alpha :I\u27f6{\mathbb{L}}^{3} is said to have constant curvature ratios (that is to say, it is a ccrcurve) if all the quotients {\epsilon}_{i}(\frac{{k}_{i+1}}{{k}_{i}}) are constant, where {\epsilon}_{i}=\u3008{V}_{i},{V}_{i}\u3009=\pm 1.
Corollary 2.2 For i=1, the ccrcurve is \frac{{\epsilon}_{1}}{{H}_{1}}.
Proof The proof can be easily seen by using the definitions of harmonic curvature and ccrcurve. □
Corollary 2.3 Let \alpha :I\u27f6{\mathbb{L}}^{3} be a ccrcurve. If \frac{{\epsilon}_{1}}{{H}_{1}}=c, c is a constant, then {(\frac{{\epsilon}_{1}}{{H}_{1}})}^{\mathrm{\prime}}=0.
Proof The proof is obvious. □