LEFSCHETZ : THEORY OF ALGEBRAIC MANIFOLDS. 9 



ing directly [M{C I + Co)] in terms of the genera [MC1C2], are 

 nil of the type (11), / being of the first degree. 



Suppose that { M (A + B) \ = \Mf{A,B)\. Clearly/ (A, B) 

 must be symmetric in A, B, this being a direct consequence of the 

 commutative law. Furthermore, 



\M{A + B-}-C) f = \Mcj>(A,B,C) f, 

 where cj) is a polynominal which, for the same reason, must be 

 symmetric in A, B, C. Let 



j{A,B) =2ai,A'BS 

 and denote by a the highest power at which either A or B occurs 

 in /. Then, since 



^{A,B,C) =2aikA'(2a„,B'^CV)\ 

 it is easily seen that mcp, A will be found at most to the power 

 // , while if r is sufficiently high, there will be a term containing 

 C with the exponent a " . 



.*. u" = a , u = 1 . 

 Thus the polynomial / must be of the first degree. As it is also 

 symmetric, we have 



j{A,B) =a{A + h^ (B + b) +c, 

 where a, b, c are certain constants still in part to be determined. 

 From this follows 



cp (A,B,C) =a(A + b) (a(B^b) (C + b) +c) ^c , 

 and the condition of symmetry will be fulfilled only if b = —c, 

 when/(A, B) =a(A + &) (B + b)-b . 



.-. \a(A'+B + b) { = \a(A^b).a{B-\-b) [- . 

 If we set a = a, ab = 3 , we have 



C = A-\-B; \ M(aC + 3) \ = j M(aA + ;^) (aB + ;^) f 

 as was to be proved. 

 Lawrence, Kan., May 30, 1916 



n 



