48 CUKTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION 
I Lace if Q, Q, Q are any three kindred Q functions with regular singular points 
0, co , 1, none of which are semisingular, relation (10) is, for these functions , 
(51) 
Q 4> (x) + Q®(x) + QQ>(x) = 0, 
where, after all common factors are divided out, <l> (x), ® (x), cj> (x), are polynomials of 
degrees not higher than the numbers given by formula, analogous to (50). 
In classes where semisingular points occur, every basis is fundamental at such a 
point, so that although two kindred families have the same connecting formulae, it is 
possible in some cases that they may have no common connecting formula in which the 
fundamental bases have exponents X', X j //, jx" ; i/, v" ; respectively. The only new 
possibility here is that in such a case we must substitute in (48) and (49), if a 'is semi- 
singular, X" + X" for X -f- X, and similarly if other points are semisingular. This merely 
increases the number given by (49) and (50) by an integer. The form of (51) is un- 
changed in that the coefficients are still polynomials, but the degrees of one or more of 
these polynomials may be larger than the limit (50) would give. But even this excep- 
tion to the general rule can occur only in a comparatively limited' range of cases under 
the classes where semisingular points enter, namely, where two of the three families 
to which the functions in (51) belong have no common connecting formula in which 
bases having exponents \', X", u', u", /, v" enter. 
i 
§ 2. EXPRESSION OF Q FUNCTIONS IN TERMS OF KINDRED P FUNCTIONS. 
We may apply the results of the preceding section in many ways. Thus, the differ- 
ential equations (38) and (39) are special cases of relation (51). Again Riemann in 
his paper, "Ueber die Flache vom kleinsten Inhalt, etc.,'" has given a formula con- 
necting a special Q function of order 2 with a kindred P function and its derivative 
Jiut an especmlly important application is the expression of Q functions in terms of 
P t.mctions and rational functions only. We shall find that no P family can be 'of 
Class 3 of Type I, and that the classes of Type IV, other than 1, 2, 3, 4, are likewise 
nnposs >.e or P families, but that in all other classes there are P families kindred to 
Z 7, ° 1 7 Fr ° m (51) We haVe ^ the0rem : A Q *»*" «*>» 
rlt rr; g, r t % two ■* tems ' but each ° f th - * -*—* 
in terms ot other P functions by means of similar formulce 
We ma y note that in the case3 ^ ^ ^.^ 
posed of elementary functions. Thus in Type I, Class 3, the family is 
com 
1 Werke, p. 324. 
