58 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
(*) If \' + // + */ = X'- A" *0, i.e., X" + // + *' = °> X' + / + /* 0, the two roots 
of (55) are - X" - / s and - X' - fT * We hardly need give here the details of a 
discussion which will be so similar to that given for Type III. The criteria are : 
X' - X" 
Class 1. A = - X" - fi' *, s £ - -r, 
" 2. A=-U 
« 3. A = - X' 
.. 
4. 4 = -X 
f 
*"*, 
^ 
*/*", 
„' = v". 
ix"s, 
^ 
= A*", 
v' * v". 
/*"* 
^ 
*A*' r , 
v' * v". 
No other classes possible. 
X' + ftf + v' *» /* - fi" * 0. The two roots of (55) are - X" - ft' * and 
X' _ n" s. The criteria are : 
Class 1. A = - X" -V «> *' * *"> ** = ""• 
« . i X'-X" 
" 3. A = - X" - p' s, X' = X", v' * v". 
'" 4. 4 = -X"-/*'a, X'^X", v'^tv". 
(d) X' + / 4- v = */ - i>" * 0. The two roots of (55) are - X' - // 5 and 
X" — /x" s. The criteria are : 
it 
n 
Class 1. A=-X" - fi" 8, X' * X", ja' = p 
" 2. A=--k l '-p»8, X' = X", /*'*/* 
" 4. ^ = - X" - ft" «, X' * X", /x' ,6 ,*" 
§ 3. EXPRESSION OF A Q FUNCTION OF ORDER 1 AS A SUM OF TWO KINDRED 
P FUNCTIONS. 
The exponent scheme of a Q family of order 1 beir. 
o 
00 1 
(61) 
« 1 ?. : 
' v' \ 
" v» X ) 
Ritter states that a function belonging to this family is in general expressible linearly 
in terms of two kindred P functions chosen from one of the following triples of P 
families : In the first triple, the three families have the same notation scheme for the 
exponents as Q, except that the first family has X'' + 1 in place of X', the second // + 1 
in place of //, and the third v' + 1 in place of •; in the second triple, there are similar 
changes in the second row of exponents in (61). This representation he does not 
I 
