CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED h;iO?T. 50 
prove to hold for all cases, but merely gives an example for a family all of whose 
points are ordinary. Our formula? of Part II, C, show that this expres>iu U is valid, 
provided (1) that we can choose two such P families kindred to Q, and (2) that the 
two P families chosen, and Q, have a common connecting formula (including the asso- 
ciated constants 0) hi which no two members of any basis hare the same cpo, it (vac pt in 
the case of equal exponents). The reason for this last condition is obvious from the 
remarks of page 48. 
The first condition is violated by Q functions of Type II, Claw 3, and of the cla> ■ - 
of Type IV other than 1, 2, 3, 4, since P families cannot belong to those classes. 
Both conditions are always satisfied for Type I; and an inspection of each cla--» for 
which P families are possible under Types II, II I, IV will show that the first con- 
dition can always be satisfied for a Q family of any of those classes. Formula (5< 
shows that the degrees of the coefficients in (51) taken for functions of the Q family 
and two kindred P families chosen as above indicated, must be zero, provided the 
second condition above is satisfied. The reader may assure himself by an examination 
of each class that this second condition is also satisfied except in the doubtful em s 
noted in the preceding section; and that here the only violation is in Class 3 of Type 
III, and Class 4 of Type IV, when the two values of A coincide, the family being then 
completely determined by its exponent scheme and apparently singular point. Hen 
the formulae of Part II, C, fail to give the desired representation, which ne vert he] 
exists. In fact if P' and P" be two functions chosen as above indicated, it is easy to 
show that constants a and /3 always exist such that the function 
which evidently has the desired exponent scheme, has also the characteristic apparently 
singular point. 
A Q, function of order 1 of any class possible for P families can alvaifs he e^nssed 
linearly in terms of kindred P functions. 
We have thus illustrated some of the difficulties that arise in a first generalization 
from P families. Our methods are sufficient for the discussion of further cases, but 
conditions become so complicated that a similar treatment of Q families of order 2, or 
of higher order, will not be attempted here. 
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