44 
Proceedings of the Royal Society of Edinburgh. [Sess. 
compatible when, and only when, a assumes a value in this sequence. 
In any particular system, the set of characteristic numbers depends, inter 
alia, upon lc{a) and l{a). 
When the given system is thus rendered compatible, there arises one of 
two cases. In the more general case, to a characteristic number there 
corresponds a solution u.(x), unique apart from a constant factor ; in this 
case the system is said to be of index of compatibility 1. On the other 
hand, when further relations between a, k(a), and 1(a) are introduced, it 
may occur that two linearly independent solutions of (I), and therefore all 
solutions of (I), satisfy conditions (T), in which case the system is said to 
be of index 2. 
Let K(o?, s) be a solution of the partial differential equation 
L,(K)-L,(K): 
Jc(x) 
0K' 
dx 
^(s)^] + W*)-^W]K = 0 
satisfying the periodic conditions 
K (x, /3)-K (x,a)^ 
f3)^Ks{x, a)J 
C^) 
( 2 ') 
for a<cr</3. Since equation (2) is symmetrical in x and s, it follows that, 
if K(cc, s) exists, it may be so chosen as to be, if not s^nnmetrical, i.e. such 
that K(s, x) = K(x, s), then at least skew-symmetrical, i.e. such that K(s, x) 
== — K(o?, s). This being so, the conditions 
K (fts) = K (a,s)'^ 
= ^ 
will also be satisfied for a^s^/3. 
Let u^(s) be a solution of 
Ls(zi) + aiU = 0, 
in which a* is a characteristic number, and let 
Then 
l(x)= K(x, s)2ii(s)ds. 
Jo. 
.^,( 1 )- l'L^(K)ui(s)ds = jLs(K)ui(s)ds 
r/3 
-f K(x, s)Ls(ui)ds 
a Ja 
R 
since the bilinear concomitant 
R = A:(s)[z^,<s)Ks(x, s)-w/(s)K(.r, s)] . . . . (3) 
vanishes between the limits of integration for all values of x in (a, ^). 
* Forsyth, Theory of Differential Equations, vol. iv, p. 252. 
