ON THE VALIDITY OF HUYGENS PRINCIPLE FOR 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS WITH 4 INDEPENDENT VARIABLES .2. A 6TH NECESSARY CONDITION

The expansion of Hadamard's necessary and sufficient condition for the validity of Huygens' principle for a second order partial differentia l equation with four independent variables is taken to fifth order. Th is expansion takes the form of a covariant Taylor series in normal coo rdinates, and is extended to a coordinate invariant expansion by impos ing invariance of the resulting coefficient under the set of trivial t ransformations. The requirement that this coefficient vanish for a Huy gens' equation provides a sixth necessary condition.