8 THE UNIVERSITY SCIENCE BULLETIN. 



usual sense for the set of the A 's, we can obtain all the genera 

 [M] as linear functions of the base genera. 



Starting from the above properties, all the formulas obtained 

 could be derived without having recourse to virtual manifolds, 

 as becomes necessary with the more specialized geometric theory 

 already discussed. 



In this connection three questions present themselves : 



(a) Is it possible to obtain other systems than that of alge- 

 braic manifolds contained in a given variety, having the proper- 

 ties outlined above? 



(b) Given a base set Ci, C2, . . . C,o, the sj'-stem defined by 

 all the C's such that /". C ==^ 2 X; C; , and a system of inte- 



gral values for the genera [C'l Ci • . . . C/* J , is there an irredu- 

 cible algebraic variety such that its hypersurfaces are in one-one 

 correspondence with the system so defined ? 



(c) Are there laws other than the one corresponding to 

 Severi's addition formula, and yielding results of interest ? 



Very little may be said at present in regard to (a) and (b), 

 but more concerning (c), which is what we proceed to do. 



8. The question must first be stated more definitely. It is 

 proposed to consider addition formulas of the type 



JM/(Ci + C2)i = iMcpiCC^) \, 

 where /, <p , are polynomials. The simplest of this type is per- 

 haps the following : 



1 Mf{C, + C,) \ = ] Mf{C,)fiC2) \ (11) 



which can be easily shown to lead to the result 



\ _ hi 

 XD=^'A,Cr, \f{D) \ =] |i|/(Ci)>. \ . 



The difficulty with laws of this general type lies in the fact 



that unless / is linear, [C1 + C2] can not be obtained readily in 



terms of the genera [Ci C2] . Restricting ourselves, therefore, to 



the case where [C1 + C2] can be obtained directly in terms of 



these genera, we propose to show that : Addition formulas yield- 



