deletions | additions       diff --git a/The_equation_for_the_ellipsoid__.tex b/The_equation_for_the_ellipsoid__.tex  index 91b92d3..538aabb 100644  --- a/The_equation_for_the_ellipsoid__.tex  +++ b/The_equation_for_the_ellipsoid__.tex 
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or equivalently,   $$y_k = \left( 1 + \frac{t}{e_k^2} \right) x_k, \qquad k=1,2,3.$$  We can formulate this as  $$\frac{e_k^2}{t+e_k^2} y_k = x_k, $$x_k(t):=\frac{e_k^2}{t+e_k^2} y_k,  \qquad k=1,2,3$$ We plug this into the expression for $E$,  $$E(t)=\frac{e_1^2 y_1^2}{(t+e_1^2)^2}+\frac{e_2^2 y_2^2}{(t+e_2^2)^2}+\frac{e_3^2 y_3^2}{(t+e_3^2)^2}-1$$ $$E(t)=\frac{x_1(t)}{e_1^2} + \frac{x_2^2(t)}{e_2^2} + \frac{x_3^2(t)}{e_3^2} - 1$$  The derivative $E'(t)$:  $$E'(t)=-\frac{2 e_1^2 y_1^2}{(t+e_1^2)^3} - \frac{2 e_2^2 y_2^2}{(t+e_2^2)^3} - \frac{2 e_3^2 y_3^2}{(t+e_3^2)^3}$$   
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