1. This lecture was filmed under the auspices of the Committee on Educational Media of the Mathematical Association of America, and an expanded version of the script has been published in The American Mathematical Monthly.
2. Kac actually discusses only the case of a single component membrane (C=1) with a convex polygonal perimeter and one or more convex polygonal holes. A he observes the result (1.7 b) follows formally by letting the polygons approach smooth curves while the extension to C>1 is obvious from his analysis.
3. This follows by iterating the defining equation Twn=μnwn.
4. It is easily shown that the trace Tr{A}=∑j−1NAjj of a matrix A is invariant under the similarity transform A′=SAS−1. On choosing S to diagonalize A′ the A′ii becomes the eigenvalues, thereby proving the result.
5. Notice that in the case of doubled or trebled edge sites bα includes only those bonds crossed as the boundary passes the site.