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...

[jajamezsz]

Let's face it, engineering math is often quite ugly. If you overthink it, it only looks back with a goofy stare..."4pi, you say? Hardihar, har, har..." In a way, it is hard to map metaphysical concepts to geometrical/algebraic rules, and one can often only use mathematical analogies to the best of their ability, "the rate of x is proportional to what is observed under certain conditions," or "assuming a uniform distribution in all directions," etc.

Anyhow, I am checking in to mention another nugget I caught upon rereading Theoretical Elements of Electrical Engineering:

"When reading the book, or using it as text-book, it is recommended:
After reading the first or general part of the present volume, to read through the first 17 chapters of 'Theory and Calculation of Alternating Current Phenomena,' omitting, however, the mathematical investigations as far as not absolutely required for the understanding of the text, and then to take up the study of the second part of the present volume, which deals with special apparatus. When reading this second part, it is recommended to parallel its study with the reading of the chapter of 'Alternating Current Phenomena' which deals with the same subject in a different manner. In this way a clear insight into the nature and behavior of apparatus will be imparted."

It's a similar progression to what was stated in my last post, quoting from "Engineering Mathematics." I have yet to see him mention where "Theory and Calculation of Electric Circuits" fits in. It would seem that Steinmetz is laying out the primer which should be enough to tackle his other works with fluency.

Thanks for tuning in!

Sincerely,
James M.