4 THE UNIVERSITY SCIENCE BULLETIN. 



mula analogous to Seven's is given by C = A-\-B; ] (aA+/i) 

 iaB-\-;3) {- = \aC-]-S\, which for a ^ (3 = 1 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'' . . . . C' is replaced by [A' B^' . . . . C"] if its degree is 

 t r, and by zero if it exceeds r. 



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



j/(A, B, . . . . C) f = [f(A, B, . . . . O] + (-ly-^jli-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 ntroduction to it. 



3. LetD = Ci + C2. Then» 



\1+Di = \ (1+C,) (J+C.) \. (1) 



The following generalizations of (1) have been obtained: 



(a) If D =.f,Ci, M = D,Do D^, then 



j M(l+D)f =^\M'\~\{1+C0\. (2) 



(b) If D = Ci-C2, ] M(l+D) \ = 



JMy^-|- = \M(1+C0 (l-hC^y'l, (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 



-)M(i+D)f = {MT7[(1+Ci)>i(. (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 



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

 nally given by Severi. 



