If I understand your setup correctly, the correct solution should be something similar to what @sebasth answered above. Specifically, the formulas should be:

R_ac = R_bc R_ac

t_ac = t_bc + R_bct_ab

Derivation:

My assumption is that you have the following formula to go from frame a to frame b:

x_b = R_abx_a + t_ab

Similarly, for frame b to frame c (same formula with subscripts appropriately replaced):

x_c = R_bcx_b + t_bc

We want to get a formula for R_ac and t_ac such that the same relationship holds true for transforming from frame a to frame c:

x_c = R_acx_a + t_ac

If you plug in R_abx_a + t_ab from the first equation for x_b in the second equation, you get

x_c = R_bc(R_abx_a + t_ab) + t_bc

x_c = R_bcR_abx_a + R_bct_ab + t_bc

Pattern matching the above with the desired final formula (x_c = R_acx_a + t_ac), we get:

x_c = (R_bcR_ab)x_a + (R_bct_ab + t_bc)

R_ac = R_bc R_ac

t_ac = t_bc + R_bct_ab