An analytical description of the y-S7 diagram is given in terms of plane vectors representing the points and lines in the diagram. This allows for a very powerful and elegant tool for deriving and using the design and synthesis properties of the y-37 diagram. Exact analytical expressions are given to derive the first-order properties of an optical system from its representation in the diagram. The Seidel (third order) aberation coefficients are expressed in terms of variables better suited to the y-Yr diagram, and the description of a computer program that implements these ideas, is included. At the request of a number of participants in the Seminar, a complete bibliography on the y-y diagram is included as an Appendix.