18 CURTISS. — BINARY FAMILIES IN A TRIPLY CONXEC TED REGION 
will always liave different groups when they belong to different classes ? This is the 
question which will now occupy us, and which will be answered affirmatively by the 
actual construction of such a classification. 
It is instructive to observe how this may be done for the c;i e n = 2. Here fami- 
lies with the same multipliers are kindred unless the two multipliers of a hole a are 
equal to each other; in this case both holes may be logarithmic for some families, and 
semisingular for others. It will readily be seen that the following is a classification of 
kind desired 
Type I. . P J * Pa ". 
Type II o'-o"S Classl - °n = °> 
lype Pa _ Pa j ciass ^ ^ ^ 
§2. THE FOUR TYPES. 
Obviously the study of connecting formula will be most essential to our work 
but it is important to notice that we need concern ourselves only with the } trt 
(12) 
yd = a y h > + y b ". 
V« =vy b f + S y 6 ", 
since the relation S c S b S a = 1, noted on page 9, shows that the generating substitution 
for c 1S expressible in terms of those for a and b. But this relation S c S b Si = 1 gives, 
besides the equation 
(13) 
Pa Pa" Pb ' Pb " pj p» = 1, 
already gl ven for the general case on page 12, an additional equation in terms of 
"' , 7f ' b ' and the multi P liers > which is most important for our subsequent 
work. We obtain this as follows : If we turn back to § 4 of Part I, A, we 
p. and Pc are roots of the invariant characteristic equation 
(14) 
that 
m x — p n 
l 
2 n 2—P 
o, 
Where U £) is the generating substitution of any basis about c ; if the inverse of 
such a generating substitution, the roots are 1 1 r, , • , , ^ • m\ 
, u« roots are 0n tlus latter hypothesis (14) 
The first 
1_ 
Pc' Pc" 
w 2 n 
but this is readily seen to be (13) in another form.* 
The other invariant is 
2 
1*1 * a - wi a n x is in fact the Drodurt TW c r* 4. <* ^ ^ 
product Det S a . Det S b . But Det S a = 9< > p », and Defc s b = pf p b ». 
