MATHEMATICS: S. LEFSCHETZ 
297 
the effect of T upon the Hnear cycles 7^ of No. 1, we find at once that all 
the w's not of the type m^j^^p^^, {/jl, p ^ p) are equal to zero, hence p' is equal 
to the number of independent forms of type. 
ivp+v - xp+vy,^ 
(3) 
which belong to 12. 
If Vp is pure p' ^ p, for otherwise 9. would possess a degenerate form (3). 
This is to be contrasted with Scorza's result 1 k ^ 2p — 1, or p ^ 2p — 1 
if 0 is pure. 
4. Assuming p' = 2 let Z, Z', be the matrices formed by the determinants 
of two forms (3) . They are both of type 
0 
A 
A' 
0 
where each square represents a matrix with p rows and columns, the matrices 
in the main diagonal having only zeroes for terms. As V is of the form 
A 
0 
0 
A' 
= a, 
(fi,v=l,2,.., 2p; a^j^^pj^^ = ap^^^j, = O), 
Vp has a complex multiplication defined by 
P p p P 
111 1 
(y, fx = 1,2, ... p), 
the (X)'s being necessarily real as they can be replaced by their conjugates. 
Finally the characteristic equation of this complex multiplication 
II — ^iiv ^11 =0, (e^j, = 0, M ^nix = 1) 
is necessarily reducible and a perfect square if Vp is pure. 
5. Let us examine the case of a real hyperelliptic surface of rank one. 
The number p' has then the value 1 or 2, if the surface is pure not elliptic. 
A fundamental period matrix corresponding to Hnear cycles forming a mini- 
mum base may be reduced to the form 
m n 
1, 0, - + ia, - + ib 
n r 
0, 1/5, - + ib, - + ic 
where m, n, r, 8, are positive integers and ac — b^ > 0. If y[, 72, . . . yip, 
are the corresponding linear cycles those of No. 1 are given by 
Ti = 7i, 72 = 72, 73 = 2y's — my[ — ndy'j, 74 = 274 — ^7i — ^'^72, 
