Steinmetz Math Typo?

[jajamezsz]

Greetings like-minded energy-rustlers,

I have been reading Steinmetz's "Theory and Calculations of Electric Circuits" and have come upon a sticking point...I am hoping to have some clarification here without going too deep into what I've already tried. The book is a free .pdf on archive.org, and if you're reading along, I'm talking about Section 23 of Chapter 2 called "Stability Curves of the Arc" and discussing the equations related to Figure 20. I can't figure out how Equation (7) comes about. I think there is a typo at the top of the page where e' should be e, instead, but even so I can't wrap my head around it. Please let me know what you think, if you get the chance. Thanks in advance to the brave genius who takes this on!

[jajamezsz]

I am on the fence about the typo. I am leaning towards the idea that it's not a typo, and that my confusion comes from the "2" in the denominator of Equation (7). Since Curve I is nonlinear, I'm pretty sure r = v/i does not apply, but rather would be defined by the slope at that point, e' (over i equals r equals 10ohms). Still, I can't seem to resolve my confusion with Equation (7).

Taking a break from this problem, I found this little gold nugget in Steinmetz's "Engineering Mathematics,"

"The following work is intended as an introduction and explanation of the mathematical side, and the most efficient method of study, appears to me, to start with 'Electrical Engineering Mathematics,' and after entering its third chapter, to take up the reading of the first section of 'Theoretical Elements,' and then parallel the study of 'Electric Engineering Mathematics,' 'Theoretical Elements of Electrical Engineering' and 'Theory and Calculation of Alternating Current Phenomena,' together with selected chapters from "Theory and Calculation of Transient Electric Phenomena' and after this, once more systematically go through all four books."

From the Lizard Wizard, himself...