deletions | additions       diff --git a/section_Equations_The_equation_for__.tex b/section_Equations_The_equation_for__.tex  index f053947..59366c5 100644  --- a/section_Equations_The_equation_for__.tex  +++ b/section_Equations_The_equation_for__.tex 
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and their derivatives are given by,  $$x_k'(t) = -\frac{e_k^2}{(t+e_k^2)^2} y_k = -\frac{x_k(t)}{t+e_k^2}, \qquad k=1,2,3.$$  The equation of the ellipsoid is reduced to a scalar equation in $t$:  $$E(t)=\frac{x_1^2(t)}{e_1^2} + \frac{x_2^2(t)}{e_2^2} + \frac{x_3^2(t)}{e_3^2} - 1$$ 1.$$  The derivative $E'(t)$ is simply:  $$E'(t)=-2\frac{x_1(t) x_1'(t)}{e_1^2} - 2\frac{x_2(t) x_2'(t)}{e_2^2} - 2\frac{x_3(t) x_3'(t)}{e_3^2}$$ x_3'(t)}{e_3^2}.$$