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We begin by realizing that $\frac{a}{c}=\frac{a}{b}\cdot \frac{b}{c}$. Thus, we have $$\frac{a}{c}=\frac{a}{b}\cdot \frac{b}{c}=\frac{\sqrt{10}}{\sqrt{21}}\cdot \frac{\sqrt{135}}{\sqrt{8}}=\sqrt{\frac{10}{21}}\cdot \sqrt{\frac{135}{8}}=\sqrt{\frac{10\cdot 135}{21\cdot 8}}.$$We can then simplify by canceling common factors under the square root. $10$ and $8$ share a factor of $2$ while $135$ and $21$ share a factor of $3$, so we have $$\sqrt{\frac{10\cdot 135}{21\cdot 8}}=\sqrt{\frac{5\cdot 45}{7\cdot 4}}.$$We now simplify further and rationalize the denominator to get: $$\sqrt{\frac{5\cdot 45}{7\cdot 4}}=\frac{15}{2\sqrt{7}}=\boxed{\frac{15\sqrt{7}}{14}}.$$