F.12 Coaxial Circles — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.12 (Coaxial Circles) (SeeligNotebooks).

Folio 0072 (Page 1 of 12): Coaxal Circles & Electromagnetics Introduction

[Folio 0072]

Folio 0072 (original scan 0072.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0072 (0072.jpg, Sheet 1 of 12):

Introduces coaxal circles via equipotentials around two charges (\(K = Q_1/Q_2 = 2\)), citing Stanley Ogilvy and Wikipedia, with radical axis \(\eta = 1\).

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0072.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 1 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0073 (Page 2 of 12): Template Table for K = 2

[Folio 0073]

Folio 0073 (original scan 0073.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0073 (0073.jpg, Sheet 2 of 12):

Sets up table columns \(\eta, R, MM'\) for \(\eta \in \{1, 1/2, 1/3, 2/3, 1/4, 3/4\}\) and their reciprocals.

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0073.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 2 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0074 (Page 3 of 12): Plan of Attack (Friday April 24, 2015)

[Folio 0074]

Folio 0074 (original scan 0074.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0074 (0074.jpg, Sheet 3 of 12):

Outlines the HP-35s COAXAL CIRCLES program, Excel verification, Kraus EMag comparison, and synthesis with harmonic conjugates, radical axes, and pole-polar duality (for Vol. II, p. 280).

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0074.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 3 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0075 (Page 4 of 12): Numerical Table for k = 1/2, 1/3, 1/4

[Folio 0075]

Folio 0075 (original scan 0075.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0075 (0075.jpg, Sheet 4 of 12):

Computes \(r-x, C, AC/BC, X, Y, Z, M'\), and \(R\) on segment \(AB\).

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0075.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 4 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0076 (Page 5 of 12): Six Landmark Points P_1..P_6 on the Equipotential Circle

[Folio 0076]

Folio 0076 (original scan 0076.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0076 (0076.jpg, Sheet 5 of 12):

Defines \(P_1..P_6\) around charges \(Q_1(-r,0)\) and \(Q_2(+r,0)\) on blue grid paper.

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0076.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 5 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0077 (Page 6 of 12): Table of Algebraic Transforms (1)–(5)

[Folio 0077]

Folio 0077 (original scan 0077.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0077 (0077.jpg, Sheet 6 of 12):

Catalogs the five fractional linear transforms relating \(k, a, x\) and \(y^2 = (2r)^2/(k^2-1)\).

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0077.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 6 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0078 (Page 7 of 12): Exact Coordinates for k = 2 (P_1, P_2, M’, R, P_3, P_4)

[Folio 0078]

Folio 0078 (original scan 0078.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0078 (0078.jpg, Sheet 7 of 12):

Derives \(P_1 = (\frac{1}{3}r, 0), P_2 = (3r, 0), R = \frac{4}{3}r, M' = \frac{5}{3}r\), and \(P_3, P_4 = (r, \pm \frac{2}{\sqrt{3}}r)\).

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0078.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 7 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0079 (Page 8 of 12): Right-Triangle Locus for P_5, P_6

[Folio 0079]

Folio 0079 (original scan 0079.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0079 (0079.jpg, Sheet 8 of 12):

Proves that right triangles \(\triangle Q_1 P_5 Q_2\) satisfy \(d_1^2 + d_2^2 = (2r)^2\), lying on the circle \(x^2 + y^2 = 2r^2\) of radius \(\sqrt{2}r\).

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0079.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 8 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0080 (Page 9 of 12): 1D Collinear Electric Field Vector Formulation

[Folio 0080]

Folio 0080 (original scan 0080.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0080 (0080.jpg, Sheet 9 of 12):

Writes \(E_i = k Q_i (x - x_i)/|x - x_i|^3\) to preserve vector sign along the line of charges.

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0080.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 9 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0081 (Page 10 of 12): Zero-Field Point Between Like Charges

[Folio 0081]

Folio 0081 (original scan 0081.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0081 (0081.jpg, Sheet 10 of 12):

Sets \(E_{\text{TOT}} = 0\) for \(+Q_1, +Q_2\) and forms the cubic/absolute-value ratio.

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0081.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 10 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0082 (Page 11 of 12): Sign Interval Analysis for x_1 < x < x_2

[Folio 0082]

Folio 0082 (original scan 0082.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0082 (0082.jpg, Sheet 11 of 12):

Tests sign cases of \(x - x_1\) and \(x - x_2\) across the three collinear intervals.

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

Editorial & Symbolic Notes

Archival scan 0082.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 11 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.

Folio 0083 (Page 12 of 12): Alternative Coordinate Setup (-d/2, +d/2)

[Folio 0083]

Folio 0083 (original scan 0083.jpg) from resources/seelig/12_coaxial-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0083 (0083.jpg, Sheet 12 of 12):

Solves for \(P_1\) with charges at \(\pm d/2\) and potential ratio \(V_1/V_2 = 1/2\).

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Editorial & Symbolic Notes

Archival scan 0083.jpg from Fred W. Seelig’s notebook dossier on F.12 Coaxial Circles (Page 12 of 12). Companion OpenDocument worksheet(s): Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes. Classical Euclidean foundations: III.35, III.36.