40 CURT1SS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
When the accessory parameters are given a set of values satisfying (41), the solutions of 
(39) constitute a Q family which is completely determined and whose symbol is 
00 1 8m 8, 
n 
/\' fi' v> 2 2 ... 2 \ 
£b*A se£ of solutions of (41) efc/foes a Q /awi'fy u«M tffo? symbol (42). Hence P families 
only are in all cases completely determined by the exponents and the regular singular 
points. 
A Q family of order n > 1 is completely determined when, and only when, we know 
in addition to its symbol (35) the values of its accessory parameters. 
We are now in a position 
between the accessory parameters of 
the families Q and Q in (34) (see p. 36). This we may easily do by remembering that 
to obtain the differential equation of Q we have only to substitute in (38) for Q the 
expression 
•(*-!)- §. 
If A x , A,,. . .,A K are the accessory parameters of Q, the accessory parameters 
Ah A», . . . , A K of Q are*" given by the equations 
(43) 4 = ^, + <r 4 [S(*-l) '+€«,] (1 = 1,2,...,*). 
■ 
- We have given at some length the work of deducing the differential equation when 
the regular singular points are at 0, co , 1 ; if they are at a, b, c, we may either use 
transformation (26) or proceed directly as in the previous case. We give here only 
the result; the equation is 
<44) ~: + 
d* Q , fl - V - X" . 1-n'- ul' . \- v '- v " •— ^ -\ d Q 
dx 1 x — a 
x — b x — c ^ x — s» J 
d x 
( r \'\"(a-b)(a-c) p' ,,{' (b - a) (b - c) v'v" (c-a)(c-b) % K B % 1 Q 
x ~ a ' x ~ h + x-c + .-Z X J Si ]( x -a)(x-b)(x * _0 ' 
the accessory parameters being designated by B t (i = 1, 2, . . . , K ). 
In this section is the answer to Klein's 1 question as to the appearance of the differ- 
ential equation of a Q family. Besides showing the complete data necessary to deter- 
mine a Q family, the differential equations here developed afford an existence proof for 
families of arbitrary order n, with an exponent scheme arbitrarily given (subject to the 
condition that the exponent sum is 1 - »). We shall also make use of this section in 
discussing the classification of Q families according to the scheme of Part I B 
1 P. 234. 
y 
