38 CURTISS. BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
• 
has, then, no logarithmic terms. Further, it is easy to see that the exponent of D' at 
is equal to or greater than X' + X" - 3 ; similarly, at oo , its exponent is as large as 
yj + p? + 3 f and at 1, as large as v + v" - 3. By a course of reasoning analogous 
to that used for the determinant D, we conclude that If is of the form 
2,*'+A"-3 (X - l)*'+""-3 (f>' ( X ), 
where <j>' (x), which may have roots at and 1, is a polynomial whose degree we 
denote by m ; the exponent of If at oo is therefore 
(\' + x" - 3 + v' + p" - 3 + m). 
But we have seen that this exponent is as large as // + //' + 3, so that we have 
m < n + 2. 
We have, therefore, for q the value 
1 
9 
4>'(*) 
X (X - 1) [JCX (X - 1) (X- *!)*! (X - S 2 Y* ... (X — 8 K ) 
°K 
But, if a-i > 1, we may easily show that $ (x) has a factor (x - stf where r > o-< 
by differentiating twice the branches 
X 
*iY i+1 [l + •{ (*-%) + .. .], 
1 + ej" (x - s,) + 
• • 
• » 
i substituting the results in If. Hence we may 
9 
1 
^r (x) 
x{x~l) lx (x -l)(x~ *!> (as - « 2 ) • • • (X - *„) 
where V («) is a polynomial of degree < « + 2. This, again, may be put into the 
form 
a?C*-l)Ls x-1 x-b x x-8* x-s 
where M l9 M 2 , M z , A l9 A 2 , . . . , A K are constant with respect to z, but are as yet 
undetermined functions of the exponents and apparently singular points. 
We may at once find M x by substituting a branch Q* in the differential equati 
of the family, thus obtaining a set of equations which must be satisfied. The first 
of these is M x = - X' X". By similarly substituting Q* an d Q">, respectively, we 
obtain the equations 
M % = ft' fi», m z = v' v". 
