BJT Differential Amplifier-Long Tailed Pair
Differential gain: 

If the base of Q1 moves down by \$-\Delta V_{be}\$, and the base of Q2 moves up by \$\Delta V_{be}\$, then the junction of \$R_1\$ and the two emitter resistors \$R_e\$ will remain fixed.  Since no signal current flows through \$R_1\$, the signal current through Q2 will be simply 

\$\frac{\Delta V_{be}\ - (-\Delta V_{be})}{2R_e}\ = \frac{V_{diff}}{2R_e}\$.

The voltage gain will then be 

\$\frac{V_o}{V_{diff}} = -\frac{R_c}{2R_e}\ \$.

Common mode gain:

The simplest way to calculate this is to note that \$R_1\$ will carry both \$I_2\$ and \$I_2\$, and these currents will be equal in magnitude.  It's therefore possible to split resistor \$R_1\$ for analysis purposes into two resistors equal to \$2R_1\$ in each leg of the pair and break the center connection.  Then from inspection the common mode gain is:

\$ \frac{V_o}{V_{CM}} = \frac{-R_c}{R_e + 2R_1}\$.

The common mode rejection ratio is the differential gain divided by the common mode gain, or: 

\$\frac{\frac{R_c}{2R_e}}{\frac{R_c}{R_e + 2R_1}}\$

or:

\$ \frac {R_e + 2R_1}{2R_e} \approx \frac{R_1}{R_e}\$.