(795 ) 
or dp, and the reduced row of signs either zero or dp, so again 
not pd. 
We can thus at all events take care that for y,, the reduced row 
of signs is not pd. 
We now continue the succession of y.’s in an arbitrary way accord- 
ing to the method given above, until after a certain finite segment 
of values a a first @ appears, for which the reduced row of signs 
changes. Then we have certainly there a value « which is in the case 
of exception, and which we call a,,. We there have a Ga,» which 
breaks up into a set of domains y, , and with each of these we 
Ui 
_may continue the succession of the y,’s. We know that the reduced row 
of signs of Gu, is not pd, and we shall show how from this ensues 
1 
that for at least one of the y, ’s the reduced row of signs is not p d. 
u o i 
As a breaking up of G, into several y,’s by contact points or 
contact ares can always be reduced to divisions into two y,’s by 
contact in a single point, we have but to show that if in the latter 
case the two composing domains y„ possess the reduced row of signs 
pd, G, must also have that same reduced row of signs. 
Let to that end in fig. 6 R be the point of contact which makes 
the division; let / be the composing domain whose boundary with 
the assumed sense of circuit passes in R from k, to k’,; then in LL 
it passes in R from k', to ky. : 
On the boundary of J two elements P, and P, can now be indi- 
cated, whose images on A’, lie between P, and P,,. 
Fig. 6. 
Likewise on the boundary of // we can choose two elements Q, 
and Q, whose images on k', lie between Q, and Q,. 
