LEFSCHETZ: THEORY OF ALGEBRAIC MANIFOLDS. 



ing directly [M (C I -i-C 2)] in terms of the genera [MC\C\], are 

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



Suppose that j M (A + S) \ = \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) \ = \McpiA,B,C) \, 

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

 symmetric in A, B, C. Let 



f{A,B) =2aikA'B^ 

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

 in f. Then, since 



cp {A,B,C) = SaiuA' (2a^„B-C")\ 

 it is easily seen that m (p, A will be found at most to the power 

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

 C with the exponent a ^ . 



:. u ^ = u , a = 1 . 

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

 symmetric, we have 



f(A,B) =a(A-^b) (5 + 6) +c, 

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

 From this follows 



cp (A, B, C) =a{A-{-h) (a(B + b) (C + 6)+c)4-c, 

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

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



.-. \ a(A + B + 6) (- = ] a(A-{-b) .a(B + b) \- . 

 If we set a = a, a6 = ^3 , we have 



C = A + B; jM(aC + /^) } = jM(aA + ;l) (aB + J) } 

 as was to be proved. 

 Lawrence, Kan., May 30, 1916 



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