1921-22.] Linear Dift'erential Systems and Integral Equations. 49 
§ 6. The Functional Relationship between Solutions of Two 
Distinct Linear Differential Equations. 
Let 
Lje(w) + a^^ -= 0 . . . . . . (13) 
and 
Ma;(r) + av = 0 ...... (13a) 
be linear difierential equations. Let K(.t, s) and k{s, x) respectively satisfy 
the partial differential equations 
L,(K)-M,(K) = 0 (14) 
M^(/d-L,(/d = 0 (14a) 
We are thus led to study the pair of simultaneous integral equations 
/i(x) - X K(x, s)v(s)ds . . . . . (15) 
v(x) — xl^ k(s, x)u(s)ds ..... (15a) 
where y is chosen so that the bilinear concomitant of Mg(?;) and K(ir, s) 
vanishes over y, and y' is similarly chosen with reference to Lg(u) and 
k(s, x). 
Let the development * of K(^c, s) be 
that of k{s, x) will be 
The systems (pr{^) and ^/^^{x), are each biorthogonal and 
normal. 
It is easily demonstrated, as in §3, that if \r is unequal to any other X, 
(pri^) and satisfy (13) and (13a) respectively. This stipulation may 
be modified when the index of one or other or of both of the systems is 
greater than unity. 
If we eliminate v or u between (15) and (15a), writing 
and 
K{x, s)= Ik (x, t)k(s, t)dt 
K(x, 8)--=^ s)k(t., x))dt, 
* This development is due to E. Schmidt, Inaugural Dissertation^ Gottingen, 1905 : 
see Lalesco, The'orie des equations integrates^ p. 96, or Goursat, Cours dJanalyse^ iii, p. 470. 
VOL. XLII. 4 
