Resistance firing circuit of thyristor
The two conditions, refers to the thyristor on and off.
When the thyristor is off, the gate-cathode voltage is

\$
V_g=\dfrac{V\cdot R}{R_1 + R}
\$

(voltage divider).  

When the thyristor is on, the resistor \$R\$ is in parallel with gate-cathode on-state resistance, wich usually is very low. Then the value of \$R\$ is negligible.
Therefore, current flows through R1, gate and cathode, and neglecting the voltage drop between gate and cathode:

\$
I_g = \dfrac{V}{R_1}
\$

The subscript \$m\$ in the book, corresponds to the maximum value of the signals.