52 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION 
we need only observe that P b ', P" have exponents // and //', respectively, and 
P/ t P c ", exponents /, v" . 
Type IV, Class 1. We easily see that the connecting formulas can always be 
taken 
P*> = £ /**"> = & P('">, 
P*") = 7 /KMO _ y ^(O. 
Having chosen P^ and P<"">, so that <7» has any preassigned value * 0, we may, by 
our choice of P ( ^, P<*">, give fi and r any values we please * 0. We complete our 
proof here as for Class 3 of Type III. Classes 2 and 3 may be discussed in the same 
way 
Type I V, Class 4. We have 
PJ = aP b ' = a, PJ, 
P a » = 8 P b " = B x P». 
Every basis is here fundamental at every hole. Taking P a ' and P a " as P*> and P< A ">, 
respectively, we may easily assure ourselves that the only possible exponents for 
Pb> Pc, are p", v" 9 and that we have always a formula 
Pi^") = y POO = 7i J*W) + ^ jxy^ 
where none of the coefficients vanish, but are otherwise arbitrary. 
The proof of our lemma being thus completed, we now proceed to the proof of the 
theorem : 
On!,, one P family with given singular points can ham a given set of exponents 
Take the given singular points at 0, «,, 1 by means of transformation (26) and let 
the g,ve„ set of exponents be X', X", j, f, „', v » f or a family p Let (pM> pM) 
fD , (iV> ' PW) ^ fundamental bases °f «* family, with the exponents indi- 
cated and suppose we are given a connecting formula in which all of them enter. 
Then by our lemma, smce families with the same exponents and singular points are 
IX: i ( "vr di : s * ction) ' a second famn * * ^ ^ -- «^ «* 
L».! ,,. ZJ. ' T '' 1S the P ° mt 0n which Mann's «-» 
proo may now be easdy applied to our more general problem. In fact we 
work 
only to follow the work of Part I A * 1 * *v f 7 ^ 
«J . m .1 T ' A ' § 7 > t0 see that here > as well as in the 
Riemann considers, the determinant 
is equal to the function 
**'+*" (x - l)"'+v" f (*), 
