56 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
(58) 
Hence if we put 8 = - X', e = - /, a constant must satisfy (57). This gives the 
condition, necessary and sufficient, that h x? (x — 1)"' be a member of the family, 
A = V (8 - 1) + V'S = - \' - fl' 8. 
Class 2. Proceeding similarly, we see that a necessary and sufficient condi- 
tion that k x x " (x — 1)"" be a member of the family is 
(59) 
A = \"(S - 1) + V"S = - \" - fl"s 
/ 
(60) 
Class 3. Here A must have both values (58) and (59) since both h 
and k x x " (x — l) v " are members of the family. A necessary result is 
_ y _ \n 
fl' - fi" 
But if (60) is satisfied, our family must be of Class 3 ; for the only solutions of 
(55) in the doubtful cases are, as we have noted, either (58) or (59); but (60) 
combined with either gives the other, so that if (60) is satisfied, our family must in 
fact have two members kx x ' (x - 1)"' and kx K " (x - 1)"", and must therefore be of 
Class 3. 
From the foregoing we easily see that the following relations are necessary and 
sufficient, for each class, that a family under the doubtful cases belong to the class 
designated: OtoW — V-** •* 
" 2. A = -\"-fi"s, s iz 
" 3. 8 
V - \" 
fi' — u 
V - X" 
X' - X" 
It 
Type III. The doubtful cases here belong under two headings characterized 
the relations *' + / + •- 0, and X' + / + • - X' - X", respectively. 
When X' + fi! + „' = 0, the two roots of (55) are - X' - /*' 8 and - X" - p" s. 
As noted in the discussion for Type II, the relation A = - X' - / * is a necessary and 
sufficient condition that k #' (x - If be a member of the family, and A - - X" - //' 8 
is a necessary and sufficient condition that kx>" (x - 1)*" be a member of the family. 
These two are, then, the only possible everywhere fundamental branches ; if a family 
is of Class 1, it has the former branch ; if of Class 2, the latter ; if of Class 3, the 
latter. The necessary and sufficient conditions for each class are : 
Class 1. A = -x'-fi' 8 , w- v -*". 
fi' — fi" 
« 2. A = - x" - fi" ,, v = X". 
" 3. ^rr-V'-p"*, X'*X». 
Since . * 0, oo , 1, we omit for Class 2 the condition . * - J±2j, Similarly herea f te r. 
