CURTISS. BINARY FAMILIES IN A TRIPLY CONNECTED Ll.GION. 31 
Since p a ' = e 2 ""', p a " = e 2 " 1 *", the difference k - k" cannot be an integer or BOO, from 
our definition of the term ordinary. 
If a is semisingular, formula (23) still holds for a fundamental basis (?/,/, y"). 
Here we have k — k" = an integer or zero, but evidently, since every basis is here 
fundamental for a, we can choose a basis for which k ^ k". 
If a is logarithmic we have 
+ 00 
(24) g a ' = (* - ay ^ gj (x - «)', gj ± 
o 
+ » 
y a " = {x- aY'^gJ' {x - a Y + 3- gj log {x - a). 
v=0 
Here k — k" = an integer or zero, but we must note the possibility of all the coeffi- 
cients g" vanishing, in which case k" has no meaning. Provided there is sonic g? * 
we suppose the notation so chosen in (2 i) that g 9 " * 0. 
In all these cases we may observe a property of the branches in a fundamental 
basis which is most important, namely that expressed by the equation 
Urn (* - a)~ k [log (x - a)]~ k y = g , g * 0, 
a 
a notation which covers all cases, if k is either or 1. When such an equation is 
satisfied, y is said to have the exponent X at a. 1 
In the ordinary case, gj and y a " have the exponents *' and *" respectively at a. 
These we rename X' and X". For a semisingular point a, gj and y a " again have the 
exponents k and k". When y n ' and y a " are so chosen that *' - k" > 0, we rename 
these exponents X' and X", respectively. When a is logarithmic, the exponent of //„' at 
a is k ; if k - k" > 0, the exponent of yj' is k", if / - k" < 0, the exponent of y a " is 
K . 
Again we rename the exponents X' and X", X' being Hie exponent of y a f and 
n 
the exponent of y". In all cases the general member of the family, c x y a + c 2 y 
have for its exponent at a but one of the two quantities X' and X". 
It will be 
peak of X' and X" as the exponents of the family at a. Let 
case of the following : A function <£ 
exponent X at the point a, if there exists some positive integer k (including k 
approaches a, being confined, however, to the neighborhood of a, we always have 
0), such that no matter how x 
lim (x - a)~ x [log (x - a)]~ k <p (r) = M, 
x = a 
M 
is some constant * 0. This seemingly arbitrary definition is nevertheless one which applies to all 
solution^ of homogeneous linear differential equations of any order at a regular point. Student* of that subj t 
will readily satisfy themselves that X is one of the exponents of a in the sense in which that term is used for such 
differential equations. 
