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I am looking for a determinant for a second order equation so that I can build a nomogram. The equation is simply: $$ x^{2} +2 a x-c = 0 $$

It can also be written in another format (which is more helpful to me), but I am not sure if it can be done in this format: $$ x^{2} +2 a x- \frac{D A^{2}}{D_{0} } = 0 $$

I have looked at another question asked here but I have not been able to apply that logic to these equations.

Thanks in advance folks, any input is appreciated!

user326210
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T. Calil
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1 Answers1

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Here is one:

$$\left|\matrix{-2a & 1 & 1 \\ -c & 0 & 1 \\ \frac{x^2}{x-1} & \frac{x}{x-1} & 1}\right|=0$$

The corresponding determinant equation and nomogram for $w^2 + uw + v = 0$ is treated in this article of mine:

The Lost Art of Nomography

Since the $-c$ scale is linear, you can use N-chart nomogram blocks to calculate $DA^2/D_0$ (either one block if $D_0$ is a constant, or two consecutive blocks if $D_0$ is another variable). The N-chart provides multiplication or division with linear scales on each side, so the existing $-c$ scale is replaced by the final scale of the N-chart combo.

Ron

Jean Marie
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Ron D.
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  • Thanks Ron, I have been reading over that article for weeks now! I am a commercial pilot working on a slide ruler for flight planning and really needed that. I will happily share the results with you once this project is completed. – T. Calil Oct 29 '15 at 11:55
  • The link to this excellent article looks broken. See instead here. – Jean Marie Dec 22 '24 at 09:33