deletions | additions       diff --git a/The_equation_for_the_ellipsoid__.tex b/The_equation_for_the_ellipsoid__.tex  index ffbc3c2..497bb1e 100644  --- a/The_equation_for_the_ellipsoid__.tex  +++ b/The_equation_for_the_ellipsoid__.tex 
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We can formulate this as  $$\frac{e_k^2}{t+e_k^2} y_k = x_k, \qquad k=1,2,3$$  We plug this into the expression for $E$,  $$E(t)=\frac{y_1^2}{(t+e_1^2)^2}+\frac{y_2^2}{(t+e_2^2)^2}+\frac{y_3^2}{(t+e_3^2)^2}-1$$ $$E(t)=\frac{e_1^2 y_1^2}{(t+e_1^2)^2}+\frac{e_2^2 y_2^2}{(t+e_2^2)^2}+\frac{e_3^2 y_3^2}{(t+e_3^2)^2}-1$$  The derivative $E'(t)$:  $$E'(t)=-\frac{2 y_1^2}{(t+e_1^2)^3} - \frac{2 y_2^2}{(t+e_2^2)^3} - \frac{2 y_3^2}{(t+e_3^2)^3}$$   
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