LEFSCHETZ: THEORY OF ALGEBRAIC MANIFOLDS. 7 



From this follows immediately the fundamental formula already 

 referred to; for if 



XjA=.f^X\Ci, U = l,2, s) 



we have 





i=l i=l 



which is (15) of the paper already quoted, but in a different 

 form. If at the right we wished, as there done, to have the sign 

 [ ] in place of j (■ , we would have to add an integer N. 

 To obtain its value in terms of the X 's we remark that accord- 

 ing to § 2, N can be expressed as a polynomial in the ratios /J^ . 



>> 

 We may therefore make >.j = 1, for all values of j , and find N 



somewhat as done loc. cit. § 4. This calculation will be omitted 

 here. 



7. The theory underlying the preceding discussion may be ab- 

 stractly stated thus: Let Ai, Ao, . . . . be a finite or infinite set 

 of magnitudes such that the sum 'S./.^A;, where the X's are 

 arbitrary integers, defines a quantity of the set. This quantity 

 is supposed to remain unchanged when the A's are arbitrarily 

 permuted together with their coefficients; that is, the commu- 

 tative law of addition holds. Furthermore, the associative law is 

 also assumed. Besides the above quantities, we consider their 

 products in finite number M = A1A2 . . . . A,. If r is an arbi- 

 trary but fixed positive integer, the number r — s = d is called the 

 dimensionality of M . We then define a function of M called its 

 genus [M] by the following properties: 



(a) [Ml is uniquely determined when M is known. 



(b) With the notations defined previously, 



B = A, + A2; \M (l-\-B)\ = \M{1+A,) {l+A^) \. 

 These two conditions suffice to make certain that if the numbers 

 [A\ A2 . . . . As ] are known we can obtain [Dj D^ . . • • D^] , 



where XjDj = .2 XJAi. In particular if there is a base in the 



