# Can you find the transfer function of two cascaded networks if you know their individual transfer functions?

If you have network N1 with transfer function H1 and network N2 with transfer function H2, is there a way to find the transfer function H3 for the network produced when cascading N1 and N2 (and assuming you cant see inside N1 or N2)?

At first I thought you could just multiply H1 and H2, but that doesn't seem to give me a right answer.
If we consider a simple LPF:

The transfer function is: $H(s) = \left(\dfrac{1}{sC_{1}R_{1} + 1}\right)$

Then if we cascade it with another LPF:

The transfer function is: $H(s) = \left(\dfrac{1}{s^2C_{1}R_{1}C_{2}R_{2} + sC_{1}R_{1} + sC_{2}R_{2} + sC_{2}R_{1} +1}\right)$

which is not the same as $\left(\dfrac{1}{sC_{1}R_{1} + 1}\right)$$\left(\dfrac{1}{sC_{2}R_{2} + 1}\right)$

Is there a shortcut to get to H3 from H1 and H2 or would I just have to calculate H3 the long way using KCL or KVL? What about if I know $R_1=R_2$ and $C_1=C_2$?