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\title{Notes for Reading JCM {}``A Dynamical Theory of the Electromagnetic
Field''}


\author{Jerry Bolton, Jr.}

\maketitle
\tableofcontents{}

\begin{abstract}
This is a collection of notes that is meant to supplement the JCM
text. It contains additional mathetmatical and modeling notes which
show how JCM arrived at various conclusions, and how some equations
and solutions are derived or found mathematically.
\end{abstract}

\section{Introduction}

This paper provides supplemental information useful to readers of
James Clerk Maxwell's {}``A Dynamical Theory of the Electromagnetic
Field.'' It contains diagrams and mathematical notes which show the
steps for deriving JCM's solutions, which were omitted from his short
paper. It also contains a number of alleged errata, prima rily textual,
in the original JCM document. This paper is organized sequentially
by JCM's original article number, and the author's notes are dated
for each article.


\section{Notes and Errata}


\subsection{Article (24) / 20010908}

\begin{description}
\item [\includegraphics{Art24}]a body $C$ connected to driving points $A$ and $B,$with forces
$X$ and $Y$ acting on these points.
\end{description}
The velocites of $A,$$B,$ and $C:$

\[
u=V_{A}\]


\[
v=V_{B}\]


\[
w=V_{C}\]


The velocity of $C$ as a function of the velocities of $A$ and $B$:

\[
w=p\cdot u+q\cdot v\]


\[
\frac{d}{dt}w=p\cdot\frac{d}{dt}u+q\cdot\frac{d}{dt}v\]


The relation of the forces to the simultaneous displacement by the
{}``general equation of dynamics'':

\[
C\cdot\left(\frac{d}{dt}w\cdot\delta z\right)=X\cdot\delta x+Y\cdot\delta y\]


The simultaneous displacement of C, A and B:

\[
\delta z=p\cdot\delta x+p\cdot\delta y\]


Solving for X and Y, we have:

\[
X=\frac{\left(C\cdot\frac{d}{dt}w\cdot\delta z-Y\cdot\delta y\right)}{\delta x}\]


\[
Y=\frac{\left(C\cdot\frac{d}{dt}w\cdot\delta z-X\cdot\delta x\right)}{\delta y}\]


However, Maxwell has later need for equations without dependency on
the simultaneous displacement. Therefore, we instead substitute for
$\delta z$:

\[
C\cdot\frac{d}{dt}w\cdot\left(p\cdot\delta x+q\cdot\delta y\right)=X\cdot\delta x+Y\cdot\delta y\]


Distributing we have:

\[
C\cdot\frac{d}{dt}w\cdot p\cdot\delta x+C\cdot\frac{d}{dt}w\cdot q\cdot\delta y=X\cdot\delta x+Y\cdot\delta y\]


Maxwell concludes therefore%
\footnote{David Vomlehn comments (20010919) that this is a well-accepted shortcut
and can be rigorously shown.%
} that: 

\[
C\cdot\frac{d}{dt}w\cdot p=X\]


\[
C\cdot\frac{d}{dt}w\cdot q=Y\]


We substitute for $\frac{dw}{dt}$:

\[
X=C\cdot\left(p\cdot\frac{d}{dt}u+q\cdot\frac{d}{dt}v\right)\cdot p\]


\[
Y=C\cdot\left(p\cdot\frac{d}{dt}u+q\cdot\frac{d}{dt}v\right)\cdot q\]


Maxwell factors, distributes C and applies the sum rule of differentiation:
(20010924)

\[
X=\frac{d}{dt}\left(C\cdot p^{2}\cdot u+C\cdot p\cdot q\cdot v\right)\]


\[
Y=\frac{d}{dt}\left(C\cdot p\cdot q\cdot u+C\cdot q^{2}\cdot v\right)\]


The momentum of C referred to A (as Maxwell calls it) is:

\[
\rho_{AC}=C\cdot p^{2}\cdot u+C\cdot p\cdot q\cdot v\]


The momentum of B referred to A:

\[
\rho_{AB}=C\cdot p\cdot q\cdot u+C\cdot q^{2}\cdot v\]


In a system of many such bodies:

\[
L=\sum(Cp^{2})\]


\[
M=\sum(Cpq)\]
\[
N=\sum(Cq^{2})\]


The momentum of the system $A$is therefore $(Lu+Mv)$and the momentum
of $B$is $(Mu+Nv)$, therefore: 

\[
X=\frac{d}{dt}(Lu+Mv)\]
\[
Y=\frac{d}{dt}(Mu+Nv)\]



\subsection{Article (25) / 20010908}

The force $\eta=\frac{d}{dt}Mu$follows logically because u and v
are independent. That is to say, the force $\eta$must change by this
amount to stabilize $B$when the velocity of $A$is changed by $\frac{du}{dt}$.


\subsection{Article (26) / 20010908}

Maxwell points out that $L$, $M$, and $N$are scalar functions of
the form and relative position of the circuits. Are they not quaternion
in nature? Why not?


\subsection{Article (29) / 20010908}

Maxwell states:

\[
Sy+\frac{d}{dt}(Mx+Ny)=0\]
And also:

\[
Y=\int_{0}^{t}ydt=\frac{-Mx}{S}\]
We may reach this conclusion by the following manipulations:

\[
Sy+\frac{d}{dt}Mx+\frac{d}{dt}Ny=0\]
\[
Sy+\frac{d}{dt}Ny=-\frac{d}{dt}Mx\]
\[
y(S+\frac{d}{dt}N)=-\frac{d}{dt}Mx\]
\[
y(S+0)=-\frac{d}{dt}Mx\]
\[
y=-\frac{d}{dt}\frac{Mx}{S}\]
Therefore:

\[
\int(y)dt=\int(-\frac{d}{dt}\frac{Mx}{S})dt=-\frac{M}{S}x\]
Which is precisely as Maxwell claims. Article (30) follows directly
from this.


\subsection{Article (31) / 20020704}

Maxwell writes: {}``multiply (1) by x and (2) by y, and add.'' He
means: {}``multiply (4) by x and (5) by y, and add.''
\end{document}
