212 
MATHEMATICS: G. M. GREEN 
Theorem II. Suppose that in the system of partial diferential 
equations 
n-1 
the derivatives y,y(.^\y(-^\ . . . ^ y ^"-^'> form a normal set. Suppose further 
that in the closed region A the coefficients a\^'^\ which are functions of the 
p independent variables, Ui, U2, . . . , Up, satisfy identically the integra- 
hility conditions 
ba 
1-1 
bu 
(^,7 = 0,1, 
dUk 
l;k, 1=1,2,. , .,p). 
Let {uf\ uf\ • . • , u^p^) he any point of A, and 3/0, yQ\ • • . , ^'o"'^^ 
he any set of n constants. Then there exists one and only one function y 
of the variables Ui, U2, . . - , Up which satisfies the system of differential 
equations, and whose derivatives y, y(^\ • • • , take on respectively 
the preassigned constant values y^, y^o\ . . 
^2 , 
yQ at the point (uf\ 
From this theorem may be inferred at once the existence of a funda- 
mental system of n solutions, yi, yz, . . . , >, such that any other solu- 
tion of the system of differential equations has the form 
y = Ciyi 4- C23'2 -f 
+ Cnyn. 
Moreover, any function of this form is a solution of the completely 
integrable system; this system is therefore a natural generalization 
of the ordinary homogeneous Hnear differential equation of the wth 
order. 
The derivatives y, y(^\ . . y("-^) which appear in the right-hand 
members of the differential equations we shall call the primary deriva- 
tives. We may now define the Wronskian of n solutions of the com- 
pletely integrable system, as the determinant formed from the primary 
derivatives of these solutions: 
(1) 
y2 
^(«-i) ^(«-i) 
71 J y2 , 
yn 
yl'' 
