The unit elongation at a certain point on the surface of a body are

measured experimentally by means of strain gages that are arranged $45$

apart $($called the $45$ strain rosette$)$ in the direction of $e_1,e_1^{\text{'}}=\frac{e_1+e_2}{\sqrt[\phantom{2}]{2}}$

and $e_2.$ If these unit elongation are designated by $a=200{10}^{-6},b=$

$50{10}^{-6},$ and $c=100{10}^{-6},$ respectively:

$($a$)$ What are the strain components ${}_{11},{}_{22}$ and ${}_{12}?$

$($b$)$ Find the principal directions, assuming ${}_{31}={}_{32}={}_{33}=0.$

$($c$)$ How will the result of part b be altered if $330.$