deletions | additions       diff --git a/The_equation_for_the_ellipsoid__.tex b/The_equation_for_the_ellipsoid__.tex  index 9171e3f..ffbc3c2 100644  --- a/The_equation_for_the_ellipsoid__.tex  +++ b/The_equation_for_the_ellipsoid__.tex 
...
$$\frac{e_k^2}{t+e_k^2} y_k = x_k, \qquad k=1,2,3$$  We plug this into the expression for $E$,  $$E(t)=\frac{y_1^2}{(t+e_1^2)^2}+\frac{y_2^2}{(t+e_2^2)^2}+\frac{y_3^2}{(t+e_3^2)^2}-1$$  The derivative of the left hand expression above: $E'(t)$:  $$E'(t)=-\frac{2 y_1^2}{(t+e_1^2)^3} - \frac{2 y_2^2}{(t+e_2^2)^3} - \frac{2 y_3^2}{(t+e_3^2)^3}$$     
...