4 THE UNIVERSITY SCIENCE BULLETIN. 



mula analogous to Seven's is given by C^A-'rB; \ ((xA-h;^) 

 (aB+/i) [- = \aC-hi3\, which for a = /:> = i yields Seven's. 



2. The symbols used will be practically those of my previous 

 paper. It will be convenient to recall them briefly. 



The genus of a manifold M will be designated by [M]. Let 

 A, B, . . . . C, be hypersurfaces of the fundamental irreducible 

 algebraic variety V^, f (A, B, . . . . C) a power series proceed- 

 ing according to positive powers of the symbols A, B, C 



Then 



(a) We will denote by [/ (A, B, . . . . C) ] the result ob- 

 tained when the constant term is left unchanged; the term in 

 A^ B^ . . . . O' is replaced by [A^ B'' . . . . C'^] if its degree is 

 "i r, and by zero if it exceeds r. 



(b) Let F ix) ^ f (x, X, . . . . X) = 4' a' x'. We will write 



j/(A, B,....C:)\= [/(A, B,.... O] + (.iy-\l(-iy a\ 



This last definition is not identical with that of my paper, but is 

 equivalent to it, as shown at the end of the Introduction to it. 

 . 3. LetZ) = Ci + C2. Then^ 



\i+D\ ^\ a+Ci) (i-fc,) f. (1) 



The following generalizations of ( 1 ) have been obtained : 



(a) If D =JiCi, M = D,D2 Dy,, then 



•j Mil+D)\ =\M'\~\{1+Q)\. (2) 



(b) If Z) = Ci-C2, ]M(1^D) } = 



|m j^^j- = \M(1+C0 (l+a)-'\, (3) 



where at the right the quotient is to be replaced by its expansion 

 in power series. 



From these formulas follows that if D = -^iCi, where the /'s 

 are any integers, then 



•jM(i+i)); = jMT;i'(2+Ci)>'(. (4) 



4. The formula to be proved will follow readily from the ex- 

 tension of (4) to the case where the /'s are any rational num- 

 bers. Suppose first that/D = C, where / is a positive integer. 

 Then I propose to show that 



\M{1+D) (. = -]M(1^C)' ( , , ^^^ 



3. The arithmetic genus. formula 1. This is slightly more general than the formula origi- 

 nally given by Severi. 



