Cyclosis in Maxwell's Electromagnetism (Vol.1)

[khil_phys]

Cyclosis in Maxwell's "Electromagnetism (Vol.1)"

On page 16, sec. 18 of his "Treatise on Electricity and Magnetism (Vol. 1)", Maxwell introduces the terms cyclosis and cyclomatic number. I cannot visualize the geometry that he describes and so the next few pages are lost on me. Please help me out with this problem.

 

[khil_phys]

Anyone?

 

[micromass]

The reason that you haven't received any replies is probably because you haven't given us much information. You might also want to show us how Maxwell defines cyclosis and what the relevant definitions and concepts are.

 

[jtbell]

Yes, you need to give us more detail, because probably very few people here (if any) have actually read Maxwell's treatise or even have a copy at hand!

 

[khil_phys]

The reason that you haven't received any replies is probably because you haven't given us much information. You might also want to show us how Maxwell defines cyclosis and what the relevant definitions and concepts are.

Okay. Here it is.

Maxwell first defines the basics of line integrals and vector potentials and introduces the del operator. Now, -dψ = Xdx + Ydy + Zdz, where ψ is the vector potential and X, Y and Z are the components of the vector R in the directions of the co-ordinate axes. Now, he writes:

"There are cases, however, in which the conditions \frac{dZ}{dy}=\frac{dY}{dz}, \frac{dX}{dz}=\frac{dZ}{dx}, \frac{dY}{dx}=\frac{dX}{dy}, which are those of Xdx + Ydy + Zdz being a complete differential, are fulfilled throughout a certain region of space, and yet the line-integral from A to P may be different for two lines, each of which lies wholly within that region. This may be the case if the region is in the form of a ring, and if the two lines from A to P pass through opposite segments of the ring. In this case, the one path cannot be transformed into the other by continuous motion without passing out of the region."
...
"Let there be p points in space, and let l lines of any form be drawn joining these points so that no two lines intersect each other, and no point is left isolated. We shall call a figure composed of lines in this way a Diagram. Of these lines, p-1 are sufficient to join the p points so as to form a connected system. Every new line completes a loop or closed path, or, as we shall call it, a Cycle. The number of independent cycles in the diagram is therefore κ = l-p+1."

"Any closed path drawn along the lines of the diagram is composed of these independent cycles, each being taken any number of times and in either direction."

"The existence of cycles is called Cyclosis, and the number of cycles in a diagram is called its Cyclomatic number."

 

[micromass]

You might want to look at graph theory. What he does reminds me very much of http://en.wikipedia.org/wiki/Planar_graph#Euler.27s_formula