Formulae with all operands within a dimension are common, even in non-OLAP systems: e.g., Profit = Sales - Expense might appear in a simple spreadsheet product. In an OLAP system, such a calculation rule would normally calculate Profit for all combinations of the other dimensions in the cube (e.g., for all Products, for all Regions, for all Time Periods, etc.) using the respective Revenue and Expense data from those same dimensions. Part of the power of an OLAP system is the extensive multi-dimensional application of such a simply stated rule, which could be specified by the OLAP application builder or created by the end user in an interactive session. The true analytical power of an OLAP server, however, is evidenced in its ability to evaluate formulae where there are members from more than one dimension. An example is a multi-dimensional allocation rule used in business unit profitability applications. If, for example, a company has a Business Unit dimension and one of the business units (XYZ) is funding a special advertising campaign for Product A, and the other business units which also sell Product A are willing to share the advertising costs in proportion to their sales of the product, then the formula would be:
ADVERTISING EXPENSE = (PRODUCT A SALES/TOTAL CORPORATION PRODUCT A SALES) * ADVERTISING EXPENSE FOR PRODUCT A FOR BUSINESS UNIT XYZ
Here, Advertising is from the Measures dimension wherever it intersects with other dimensions (e.g., Business Unit, Product), but Product A Sales is more specific; it is Sales from the Measures dimension restricted to the Product A member from the Product dimension. The Advertising Expense to be shared is the Advertising Expense for Product A spent by Business Unit XYZ that the business units which have non-zero sales of Product A agreed to share. These references to several dimensions within the same rule make it a Cross-Dimensional Formula.
from OLAP Glossary, 1995