Condition for a space curve to lie on a sphere

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Problem 3.1.7 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

Let x(s)x(s) be a curve in R3\mathbb {R}^{3} parameterized by arc-length. Assume that τ(s)≠0\tau (s) \neq 0 and k′(s)≠0k'(s) \neq 0 for all ss. Show that a necessary and sufficient condition for x(s)x(s) to lie on a sphere is that

1k2(s)+1τ2(s)⋅k′2(s)k4(s)=constant. \frac{1}{k^2(s)} + \frac{1}{\tau ^2(s)} \cdot \frac{k'^2(s)}{k^4(s)} = \text{constant}.
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References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.