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MATHEMATICS: S. LEFSCHETZ 
and in general p = p' = 1, unless there is a singular relation as defined by 
G. Humbert. If the surface is not elliptic this relation can only be of type 
\a -\- lib -\- ybc = 0, (X, ju, v, integers) (3) 
and there can only be one such relation. In this case p = p' = 2, and the 
condition of existence becomes now, assuming as we may, v > 0, 
W + p,ab - vbb^ > 0 
which assures us of the effective existence of the surface. If there are two 
singular relations such as (5) the surface is elUptic and p = p' = 3. 
In addition to (5) there may be in the non-elliptic case as well as in the 
other a singular relation independent of (5) and reducible to the form 
X (b^ — ac) + /X = 0, (X, ju, positive integers) 
and then p — p' = 1, both cases being realizable. Thus there are six dis- 
tinct types of real hypereUiptic surfaces for which p, p' have the values: (1, 1), 
(1,2), (2, 2), (2, 3), (3, 3), (3, 4), the last three corresponding to elliptic cases. 
