${a_2} = {a_1}\alpha \beta $ $ \Rightarrow \,\,\,[{L_2}T_2^{ - 2}]\, = [{L_1}T_1^{ - 2}]\,\alpha \beta $......$(ii)$

${F_2} = \frac{{{F_1}}}{{\alpha \beta }}$ $ \Rightarrow \,\,\,[{M_2}{L_2}T_2^{ - 2}]\, = \,[{M_1}{L_1}T_1^{ - 2}]\, \times \frac{1}{{\alpha \beta }}$......$(iii)$

Dividing equation $(iii)$\ by equation $(ii)$ we get ${M_2} = \frac{{{M_1}}}{{(\alpha \beta )\,\alpha \beta }}$ $ = \frac{{{M_1}}}{{{\alpha ^2}{B^2}}}$

Squaring equation $(i)$ and dividing by equation $(ii)$ we get ${L_2} = {L_1}\frac{{{\alpha ^3}}}{{{\beta ^3}}}$

Dividing equation $(i)$ by equation $(ii)$ we get ${T_2} = {T_1}\frac{\alpha }{{{\beta ^2}}}$