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as nine others of a sort excluded from the present consideration, namely perspective transformations. I t is of interest to observe that the three other inflexional triangles of each invariant cubic of the third class are themselves members of the system, and are therefore transformed into themselves, not interchanged, by the eight non-perspective collineations of the cubic. Of plane curves of higher order than the third, it is easily shown that only highly specialized classes are collinear with themselves, and that of these classes the groups are correspondingly small. I t appears possible, however, to extend this method of inquiry to such interesting topics as these : (1) What simultaneous invariant conditions must two non-singular collineations satisfy in order to belong to the group leaving a common conic unaltered ? (2) What invariant conditions are met by collineations which leave unaltered a quadric surface ? a twisted cubic? a twisted quartic curve ?

A generating function for the number of permutations with an assigned number of sequences