Q$3$: $($a$)$ Describe the triaxial compression test for the determination of the shearing resistance of soils, explaining the principles of three types of test which are commonly conducted.

$($b$)$ From a study of the Mohr circle diagram for a specimen tested in drained trinxial compression, show that at failure:

${}_1={}_3\frac{1+sin{}_d}{1-sin{}_d}+\frac{2c_d*cos{}_d}{1-sin{}_d}$

Where or and os are the major and minor principal stresses.