The well known Berkeley Physics Course - volume 2 ("Electricity
and Magnetism" by E.M. Purcell, Mcgraw-Hill Book Company,
New York, 1965) also tells readers that " a set of equations like (18)
CAN BE SOLVED FOR THE V's in terms of Q's. The B's [in
equations 19] are called potential coefficients; they could be
computed from the C's, or vice versa."
This is another example of the misconception exposed by John.
Once again I looked carefully for the term "gauge invariance"
but it was not there. What I found instead was this:
"It appears [see (18)] that the electrical behavior of this system
is characterized by nine constants C11, C12,...,C33. In fact only
six constants are necessary, for it can be proved that in any system
C12=C21, C13=C31 and C23=C32. Why this should be so is not
obvious. Problem 3.27 will suggest a proof based on conservation
of energy. ... The C's in Eqs 18 are called the coefficients of
capacitance.... The capacitance of a two-plate capacitor, is not the
same as C11 (or C22 or C33), but it is of course related to them."
The discussion leading to Equations (18) is very good, in my
opinion. The author states: "We need not keep account of the
charge on the inner surface of the surrounding shell ('infinity').
since it will always be -(Q1+Q2+Q3)."
I am puzzled by the fact that the term "matrix inversion" was
not mentioned by the author. Most students would naturally
think that the phrase "equations can be solved for V's" implies
inversion but the author did not say so. Perhaps he would
calculate B's from C's in the same way as John. Perhaps he
was simply trying to avoid nuances. In other words, was
he honestly ignorant or was he deliberately evasive?
Ludwik Kowalski